HELM Book 1.5 Formulae and substitution exercises.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": true, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Evaluate simple formula", "pickingStrategy": "random-subset", "pickQuestions": "2", "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Volume of a solid", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Given the formula for a cone or a cylinder, and values for r and h, find the volume.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The formula for the volume of a {sel_solid} is {sel_formula}.
", "advice": "The volume is given by {sel_formula} where \\(r=\\var{r}\\) cm and \\(h=\\var{h}\\) cm.
\nSo volume $V=\\var{advice} \\approx \\var{answer}$ cm$^3$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"solid": {"name": "solid", "group": "Ungrouped variables", "definition": "[\"cylinder\",\"cone\"]", "description": "", "templateType": "anything", "can_override": false}, "formula": {"name": "formula", "group": "Ungrouped variables", "definition": "[latex(\"V=\\\\pi r^2h\"),latex(\"V=\\\\frac13\\\\pi r^2h\")]", "description": "", "templateType": "anything", "can_override": false}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "", "templateType": "anything", "can_override": false}, "selection": {"name": "selection", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "sel_solid": {"name": "sel_solid", "group": "Ungrouped variables", "definition": "solid[selection]", "description": "", "templateType": "anything", "can_override": false}, "sel_formula": {"name": "sel_formula", "group": "Ungrouped variables", "definition": "formula[selection]", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "precround(eval(eval_formula[selection],[\"r\":r,\"h\":h]),0)", "description": "", "templateType": "anything", "can_override": false}, "eval_formula": {"name": "eval_formula", "group": "Ungrouped variables", "definition": "[expression(\"pi*r^2*h\"),expression(\"1/3*pi*r^2*h\")]", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "latex(['','\\\\frac{1}{3}'][selection] + '\\\\pi*' + r+'^2' + '*' + h)", "description": "latex([\"\",\"\\\\frac{1}{3}\"][selection]) + latex(\"\\\\pi\\\\times\") + latex(expression(r + \"^2\" + h))
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["solid", "formula", "r", "h", "selection", "sel_solid", "sel_formula", "answer", "eval_formula", "advice"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find \\(V\\) when \\(r=\\var{r}\\) cm and \\(h=\\var{h}\\) cm. Give your answer correct to \\(0\\) decimal places.
\n\\(V=\\)[[0]] cm\\(^3\\)
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "answer", "maxValue": "answer", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": 0, "precisionPartialCredit": "75", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Evaluate linear function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "evaluate the function y=ax+b for given values of a,b and x, each of which may be positive or negative.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "For the formula \\(\\simplify{y={a}x+{b}}\\), find the value of \\(y\\) when \\(x=\\var{x}\\).
", "advice": "$y = \\var{a}\\times(\\var{x})+\\var{b} = \\var{a*x} + \\var{b} = \\var{a*x+b}$
\n$y = \\var{a}\\times(\\var{x})\\var{b} = \\var{a*x} \\var{b} = \\var{a*x+b}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)*random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "x"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a*x+b", "maxValue": "a*x+b", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Evaluate formula with division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Compute P = 3/(QR) given values for Q and R.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If \\(\\displaystyle{P=\\frac{3}{QR}}\\), find \\(P\\) if \\(Q=\\var{Q}\\) and \\(R=\\var{R}00\\).
", "advice": "$P=\\dfrac{3}{\\var{Q}\\times\\var{R}} = \\dfrac{3}{\\var{Q*R}} \\approx \\var{precround(3/(Q*R),3)} $
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Q": {"name": "Q", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "random(0.100..0.900 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "3/(Q*R)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Q", "R", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "precround(3/({Q}*{R}),3)", "maxValue": "precround(3/({Q}*{R}),3)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Simple quadratic evaluation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Calculate R = ap^2 for given positive values of a and p.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If \\(R=\\var{a}p^2\\), find \\(R\\) when \\(p=\\var{p}\\)
", "advice": "$R = \\var{a}\\times\\var{p}^2 = \\var{a}\\times\\var{p*p} = \\var{a*p*p}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(5..19)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "p"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a*p^2", "maxValue": "a*p^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Further evaluation", "pickingStrategy": "random-subset", "pickQuestions": "2", "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Evaluate formula with multiple operations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Evaluate pi / (2r+s) given values for r and s (r>0, s positive or negative)
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If \\(\\displaystyle{M=\\frac{\\pi}{2r+s}}\\), find \\(M\\) if \\(r=\\var{r}00\\) and \\(s=\\var{s}\\).
", "advice": "$M=\\dfrac{\\pi}{2\\times\\var{r}+\\var{s}} = \\dfrac{\\pi}{\\var{2*r+s}}\\approx\\var{precround(pi/(2*r+s),3)}$
\n$M=\\dfrac{\\pi}{2\\times\\var{r}\\var{s}} = \\dfrac{\\pi}{\\var{2*r+s}}\\approx\\var{precround(pi/(2*r+s),3)}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"r": {"name": "r", "group": "Ungrouped variables", "definition": "random(10.1..29.9 #0.2)", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(-0.9..0.9 #0.1 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["r", "s"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "pi/(2*r+s)", "maxValue": "pi/(2*r+s)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Evaluate formula with square root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "evaluate sqrt(x/z) where x and z are random positive decimals.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If \\(\\displaystyle{y=\\sqrt{\\frac{x}{z}}}\\), find \\(y\\) if \\(x=\\var{x}\\) and \\(z=\\var{z}\\). Give your answer correct to 3 decimal places.
", "advice": "$\\displaystyle{\\sqrt{\\frac{\\var{x}}{\\var{z}}}\\approx \\sqrt{\\var{precround(x/z,5)}}}\\approx \\var{precround(sqrt(x/z),3)}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"z": {"name": "z", "group": "Ungrouped variables", "definition": "precround(random(10.1..19.9 #0.1),3)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "precround(random(10.100..19.900 #0.1),3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["z", "x"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sqrt(x/z)", "maxValue": "sqrt(x/z)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Moment of inertia", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Evaluate J = (1/2)Ma^2 for random values of M and a.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The moment of inertia of an object is a measure of its resistance to rotation. It depends upon both the mass of the object and the distribution of mass about the axis of rotation. It can be shown that the moment of inertia, \\(J\\), of a solid disc rotating about an axis through its centre and perpendicular to the plane of the disc, is given by the formula \\[J=\\frac12Ma^2\\] where \\(M\\) is the mass of the disc and \\(a\\) is its radius.
", "advice": "$J=\\frac12Ma^2 = \\frac12\\times\\var{(M)}\\times\\var{(a)}^2 = \\var{(0.5*M*a*a)}$ kg.m$^2$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"M": {"name": "M", "group": "Ungrouped variables", "definition": "random(5..20)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3..15 except M)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["M", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find the moment of inertia of a disk of mass \\(\\var{M}\\) kg and radius \\(\\var{a}\\) m.
\n\\(J=\\)[[0]] kg m\\(^2\\)
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "moment of inertia", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.5*M*a^2", "maxValue": "0.5*M*a^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Formula - 4 variables", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "evaluate a function (4a)/(pi*b^2cd) given random values for a,b,c,d.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If \\(\\displaystyle{\\eta=\\frac{4Q_P}{\\pi d^2Ln}}\\)evaluate \\(\\eta\\) when \\(Q_P=\\var{Q_P}\\), \\(d=\\var{d}\\), \\(L=\\var{L}\\) and \\(n=\\var{n}\\).
", "advice": "$\\begin{align}\\eta &= \\dfrac{4Q_P}{\\pi d^2Ln}\\\\ &= \\dfrac{4\\times\\var{Q_P}}{\\pi\\times\\var{d}^2\\times\\var{L}\\times\\var{n}}\\\\&=\\dfrac{4\\times\\var{Q_P}}{\\pi\\times\\var{d^2}\\times\\var{L}\\times\\var{n}}\\\\&=\\dfrac{var{4*Q_P}}{\\var{pi*d*d*L*n}}\\\\&\\approx\\var{precround(4*Q_P/(pi*d*d*L*n),4)}\\end{align}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Q_P": {"name": "Q_P", "group": "Ungrouped variables", "definition": "random(0.0001..0.0009 #0.0001)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(0.01..0.09 #0.01)", "description": "", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "Ungrouped variables", "definition": "random(0.1..0.9 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Q_P", "d", "L", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "4*Q_P/(pi*d^2*L*n)", "maxValue": "4*Q_P/(pi*d^2*L*n)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Metric conversion", "pickingStrategy": "random-subset", "pickQuestions": "2", "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Metres to cm", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "convert a random length in m to centimetres
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "To convert a length measured in metres (m) to one measured in centimetres (cm), the length in metres is multiplied by \\(100\\).
", "advice": "$\\var{m}$ metres = $\\var{m}\\times 100 = \\var{m*100}$ cm
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(0.1..60 # 0.1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Convert a length of \\(\\var{m}\\) m to cm.
\n[[0]] cm
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "cm", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "100*m", "maxValue": "100*m", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Square metres to square cm", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Convert a random value in square metres to square centimetres.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "To convert an area measured in square metres (m\\(^2\\)) to one measured in square centimetres (cm\\(^2\\)), the area in square metres is multiplied by \\(10 000\\) or \\(10^4\\).
", "advice": "$\\var{m}$ m$^2 = \\var{m}\\times 10000 = \\var{m*10000}$ cm$^2$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(0.1..10 # 0.01)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Convert an area of \\(\\var{m}\\) m\\(^2\\) to cm\\(^2\\).
\n[[0]] cm\\(^2\\)
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "cm\\(^2\\)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "10000*m", "maxValue": "10000*m", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Cubic metres to cubic cm", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Convert a random number of cubic metres into cubic centimetres
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "To convert a volume measured in cubic metres (m\\(^3\\)) to one measured in cubic centimetres (cm\\(^3\\)), the volume in cubic metres is multiplied by \\(1 000 000\\) or \\(10^6\\).
", "advice": "$\\var{m}$ m$^3 = \\var{m}\\times 1000000 = \\var{m*1000000}$ cm$^3$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "precround(random(0.1..10 #0.01),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Convert a volume of \\(\\var{m}\\) m\\(^3\\) to cm\\(^3\\).
\n[[0]] cm\\(^3\\)
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "cm\\(^3\\)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000000*m", "maxValue": "1000000*m", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Basic transposition", "pickingStrategy": "random-subset", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "Transpose linear formula", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Rearrange a linear function in x and y to make y the subject. Line variables are randomised.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Transpose the formula \\(\\simplify{{selected_formula}}\\) to make \\(\\var{selected_variable}\\) the subject.
", "advice": "$\\begin{align}\\var{simplify(expression(a+\"x+\"+b+\"y\"),[\"basic\",\"unitFactor\"])}&=\\var{c}\\\\ \\var{b}y&=\\var{simplify(expression(c+\"-\"+a+\"x\"),[\"basic\",\"unitFactor\"])}\\qquad\\textrm{move }\\var{a}x\\textrm{ to the right}\\\\ y&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+b+\"-\"+a+\"/\"+b+\"x\"),[\"basic\",\"unitFactor\",\"unitDenominator\"])}\\qquad\\textrm{Divide by }\\var{b}\\\\y&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+b+\"-\"+a+\"/\"+b+\"x\"),\"all\")}\\end{align}$
\n$\\begin{align}\\var{simplify(expression(a+\"x+\"+b+\"y\"),[\"basic\",\"unitFactor\"])}&=\\var{c}\\\\ \\var{a}x &= \\var{simplify(expression(c+\"-\"+b+\"y\"),[\"basic\",\"unitFactor\"])}\\qquad\\textrm{move }\\var{b}y\\textrm{ to the right}\\\\ x&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+a+\"-\"+b+\"/\"+a+\"y\"),[\"basic\",\"unitFactor\",\"unitDenominator\"])}\\qquad\\textrm{Divide by }\\var{a}\\\\x&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+a+\"-\"+b+\"/\"+a+\"y\"),[\"all\"])}\\end{align}$
\n$\\begin{align}\\var{simplify(expression(a+\"x+\"+b+\"y-\"+c),[\"basic\",\"unitFactor\"])}&=0\\\\ \\var{simplify(expression(a+\"x+\"+b+\"y\"),[\"basic\",\"unitFactor\"])}&=\\var{c}\\qquad\\textrm{move }\\var{c}\\\\ \\var{b}y&=\\var{simplify(expression(c+\"-\"+a+\"x\"),[\"basic\",\"unitFactor\"])}\\qquad\\textrm{move }\\var{a}x\\textrm{ to the right}\\\\ y&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+b+\"-\"+a+\"/\"+b+\"x\"),[\"basic\",\"unitFactor\",\"unitDenominator\"])}\\qquad\\textrm{Divide by }\\var{b}\\\\y&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+b+\"-\"+a+\"/\"+b+\"x\"),\"all\")}\\end{align}$
\n$\\begin{align}\\var{simplify(expression(a+\"x+\"+b+\"y-\"+c),[\"basic\",\"unitFactor\"])}&=0\\\\ \\var{simplify(expression(a+\"x+\"+b+\"y\"),[\"basic\",\"unitFactor\"])}&=\\var{c}\\qquad\\textrm{move }\\var{c}\\\\ \\var{a}x &= \\var{simplify(expression(c+\"-\"+b+\"y\"),[\"basic\",\"unitFactor\"])}\\qquad\\textrm{move }\\var{b}y\\textrm{ to the right}\\\\ x&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+a+\"-\"+b+\"/\"+a+\"y\"),[\"basic\",\"unitFactor\",\"unitDenominator\"])}\\qquad\\textrm{Divide by }\\var{a}\\\\x&=\\var[fractionNumbers]{simplify(expression(c+\"/\"+a+\"-\"+b+\"/\"+a+\"y\"),[\"all\"])}\\end{align}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "formula": {"name": "formula", "group": "Ungrouped variables", "definition": "[substitute([\"a\":a,\"b\":b,\"c\":c],expression(\"a*x+b*y=c\")),\n substitute([\"a\":a,\"b\":b,\"c\":c],expression(\"a*x+b*y-c=0\"))]", "description": "", "templateType": "anything", "can_override": false}, "sel_formula": {"name": "sel_formula", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "selected_formula": {"name": "selected_formula", "group": "Ungrouped variables", "definition": "formula[sel_formula]", "description": "", "templateType": "anything", "can_override": false}, "variables": {"name": "variables", "group": "Ungrouped variables", "definition": "[latex(\"x\"),latex(\"y\")]", "description": "", "templateType": "anything", "can_override": false}, "sel_variable": {"name": "sel_variable", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "selected_variable": {"name": "selected_variable", "group": "Ungrouped variables", "definition": "variables[sel_variable]", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "if(sel_variable=0,substitute([\"a\":a,\"b\":b,\"c\":c],expression(\"1/a*(c-b*y)\")),substitute([\"a\":a,\"b\":b,\"c\":c],expression(\"1/b*(c-a*x)\")))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "formula", "sel_formula", "selected_formula", "variables", "sel_variable", "selected_variable", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\(\\var{selected_variable}=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{answer}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Transpose Gas Formula", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Transpose PV=RT to make a random variable the subject.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Transpose the formula \\(PV=RT\\) to make \\(\\var{select_subject}\\) the subject.
", "advice": "To make P V R T the subject, you need to divide both sides by V P T R
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"subjects": {"name": "subjects", "group": "Ungrouped variables", "definition": "[latex(\"P\"),latex(\"V\"), latex(\"R\"), latex(\"T\")]", "description": "", "templateType": "anything", "can_override": false}, "select": {"name": "select", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything", "can_override": false}, "select_subject": {"name": "select_subject", "group": "Ungrouped variables", "definition": "subjects[select]", "description": "", "templateType": "anything", "can_override": false}, "solutions": {"name": "solutions", "group": "Ungrouped variables", "definition": "[expression(\"RT/V\"), expression(\"RT/P\"), expression(\"PV/T\"),expression(\"PV/R\")]", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "solutions[select]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["subjects", "select", "select_subject", "solutions", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\(\\var{select_subject}=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{answer}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Harder transposition", "pickingStrategy": "random-subset", "pickQuestions": "2", "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Transpose linear formula 3 variables", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Rearrange a linear formula au + bv + cw = d to make one of u,v,w the subject.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Transpose the formula \\(\\simplify{{formula}}\\) to make \\(\\var{selected_variable}\\) the subject.
", "advice": "$\\begin{align}\\var{simplify(expression(a+\"*u+\"+b+\"*v+\"+c+\"*w\"),\"basic\")}=\\var{d}\\\\ \\var{a}u&=\\var{simplify(expression(d+\"-\"+b+\"*v-\"+c+\"*w\"),\"basic\")}\\\\u&= \\var[fractionNumbers]{simplify(expression(\"(\"+d+\"-\"+b+\"*v-\"+c+\"*w)/\"+a),\"all\")}\\\\ u &= \\var{simplify(expression({d}+\"/\"+{a}+\"-\"+{b}+\"/\"+{a}+\"*v-\"+{c}+\"/\"+{a}+\"*w\"),\"all\")} \\end{align}$
\n$\\begin{align}\\var{simplify(expression(a+\"*u+\"+b+\"*v+\"+c+\"*w\"),\"basic\")}=\\var{d}\\\\ \\var{b}v&=\\var{simplify(expression(d+\"-\"+a+\"*u-\"+c+\"*w\"),\"basic\")}\\\\v&= \\var[fractionNumbers]{simplify(expression(\"(\"+d+\"-\"+a+\"*u-\"+c+\"*w)/\"+b),\"all\")}\\\\ v &= \\var{simplify(expression({d}+\"/\"+{b}+\"-\"+{a}+\"/\"+{b}+\"*u-\"+{c}+\"/\"+{b}+\"*w\"),\"all\")} \\end{align}$
\n$\\begin{align}\\var{simplify(expression(a+\"*u+\"+b+\"*v+\"+c+\"*w\"),\"basic\")}=\\var{d}\\\\ \\var{c}w&=\\var{simplify(expression(d+\"-\"+a+\"*u-\"+b+\"*v\"),\"basic\")}\\\\w&= \\var[fractionNumbers]{simplify(expression(\"(\"+d+\"-\"+a+\"*u-\"+b+\"*v)/\"+c),\"basic\")}\\\\ w &= \\var{simplify(expression({d}+\"/\"+{c}+\"-\"+{a}+\"/\"+{c}+\"*u-\"+{b}+\"/\"+{c}+\"*v\"),\"all\")} \\end{align}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "formula": {"name": "formula", "group": "Ungrouped variables", "definition": "substitute([\"a\":a,\"b\":b,\"c\":c, \"d\":d],expression(\"a*u+b*v+c*w=d\"))\n ", "description": "", "templateType": "anything", "can_override": false}, "variables": {"name": "variables", "group": "Ungrouped variables", "definition": "[latex(\"u\"),latex(\"v\"), latex(\"w\")]", "description": "", "templateType": "anything", "can_override": false}, "sel_variable": {"name": "sel_variable", "group": "Ungrouped variables", "definition": "random(0,1,2)", "description": "", "templateType": "anything", "can_override": false}, "selected_variable": {"name": "selected_variable", "group": "Ungrouped variables", "definition": "variables[sel_variable]", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "if(sel_variable=0,substitute([\"a\":a,\"b\":b,\"c\":c,\"d\":d],expression(\"1/a*(d-b*v-c*w)\")),\n if(sel_variable=1, substitute([\"a\":a,\"b\":b,\"c\":c,\"d\":d],expression(\"1/b*(d-a*u-c*w)\")),\n substitute([\"a\":a,\"b\":b,\"c\":c,\"d\":d],expression(\"1/c*(d-a*u-b*v)\"))))", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "formula", "variables", "sel_variable", "selected_variable", "answer", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\(\\var{selected_variable}=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{answer}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Transpose formula with square root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Rearrange a formula with a square root to make a variable under the root the subject.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Transpose the formula \\(v=\\sqrt{\\simplify{x+{a}*y}}\\) to make \\(\\var{selected_variable}\\) the subject.
", "advice": "$\\begin{align}v&=\\sqrt{\\simplify{x+{a}*y}}\\\\ v^2 &= \\simplify{x+{a}*y}\\qquad\\text{square both sides}\\\\ \\simplify{v^2-{a}y} &=x\\qquad\\text{move }\\var{a}y\\text{ to the other side}\\\\x&= \\simplify{v^2-{a}y} \\end{align}$
\n$\\begin{align}v&=\\sqrt{\\simplify{x+{a}*y}}\\\\ v^2 &= \\simplify{x+{a}*y}\\qquad\\text{square both sides}\\\\ \\simplify{v^2-x} &=\\var{a}y \\qquad\\text{move }x\\text{ to the other side}\\\\ \\simplify{(v^2-x)/{a}} &= y\\qquad\\text{divide by }\\var{a}\\\\y&=\\simplify{(v^2-x)/{a}} \\end{align}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "variables": {"name": "variables", "group": "Ungrouped variables", "definition": "[latex(\"x\"),latex(\"y\")]", "description": "", "templateType": "anything", "can_override": false}, "sel_variable": {"name": "sel_variable", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "selected_variable": {"name": "selected_variable", "group": "Ungrouped variables", "definition": "variables[sel_variable]", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "if(sel_variable=0, substitute([\"a\":a],expression(\"v^2-a*y\")), substitute([\"a\":a],expression(\"1/a*(v^2-x)\")))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "variables", "sel_variable", "selected_variable", "answer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\(\\var{selected_variable}=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{answer}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Transpose rational formula", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Choice of 2 formulae. The first is a fraction of the form y=(r+x)(1-rx). The second is of the form y=sqrt[(1-x)/(1+x) ]. Rearrange to make x the subject.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Make \\(x\\) the subject of \\(\\displaystyle{y=\\var{formula[select]}}\\)
", "advice": "$\\begin{align}y&=\\frac{r+x}{1-rx}\\\\ y(1-rx) &= r+x\\qquad\\text{multiply both sides by }(1-rx)\\\\y-ryx &= r+x\\qquad\\textrm{expand LHS}\\\\y-r &=x+ryx\\qquad\\text{rearrange}\\\\y-r&=x(1+ry)\\qquad\\text{factorise RHS}\\\\\\frac{y-r}{1+ry}&=x\\qquad\\text{divide by }(1+ry)\\\\x&=\\frac{y-r}{1+ry} \\end{align}$
\n$\\begin{align} y&=\\sqrt{\\frac{x-1}{x+1}}\\\\y^2&=\\frac{x-1}{x+1}\\qquad\\text{square}\\\\y^2(x+1)&=x-1\\qquad\\text{multiply by }(x+1)\\\\y^2x+y^2&=x-1\\qquad\\text{expand LHS}\\\\y^2x-x&=-y^2-1\\qquad\\text{rearrange}\\\\x(y^2-1)&=-y^2-1\\qquad\\text{factorise LHS}\\\\x&=\\frac{-y^2-1}{y^2-1} \\end{align}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"formula": {"name": "formula", "group": "Ungrouped variables", "definition": "[latex(\"\\\\frac\\{r+x\\}\\{1-rx\\}\"),latex(\"\\\\sqrt\\{\\\\frac\\{x-1\\}\\{x+1\\}\\}\")]", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "[expression(\"(y-r)/(1+y*r)\"),expression(\"(1+y^2)/(1-y^2)\")]", "description": "", "templateType": "anything", "can_override": false}, "select": {"name": "select", "group": "Ungrouped variables", "definition": "random(0..length(formula)-1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["formula", "answer", "select"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\(x=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{answer[select]}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Transpose complex formula", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Rearrange a complex formula involving squares, square roots, fractions and additions. This is a fixed question with no randomisation.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Transpose \\(\\displaystyle{q=A_1\\sqrt{\\frac{2gh}{\\left(A_1/A_2\\right)^2-1}}}\\) for \\(A_2\\).
\n(Note: to enter the variable name \\(A_1\\) type A_1, for \\(\\sqrt{a/b}\\) type sqrt(a/b).)
$\\begin{align} q&=A_1\\sqrt{\\frac{2gh}{\\left(A_1/A_2\\right)^2-1}}\\\\ \\frac{q}{A_1}&=\\sqrt{\\frac{2gh}{\\left(A_1/A_2\\right)^2-1}}\\qquad\\textrm{divide by }A_1\\\\ \\left(\\frac{q}{A_1}\\right)^2&=\\frac{2gh}{\\left(A_1/A_2\\right)^2-1}\\qquad\\textrm{square}\\\\\\left(\\frac{q}{A_1}\\right)^2\\left(\\left(\\frac{A_1}{A_2}\\right)^2-1\\right)&=2gh\\qquad\\textrm{multiply by the LH denominator}\\\\\\left(\\left(\\frac{A_1}{A_2}\\right)^2-1\\right)&=2gh\\left(\\frac{A_1}{q}\\right)^2\\qquad\\textrm{divide by }\\left(\\frac{q}{A_1}\\right)^2\\\\ \\left(\\frac{A_1}{A_2}\\right)^2&=2gh\\left(\\frac{A_1}{q}\\right)^2+1\\qquad\\textrm{add 1 to both sides}\\\\ A_1^2 &= \\left[ 2gh\\left(\\frac{A_1}{q}\\right)^2+1 \\right]A_2^2\\qquad\\textrm{multiply by }A_2^2\\\\A_2^2&=\\frac{A_1^2}{2gh\\left(\\frac{A_1}{q}\\right)^2+1}\\qquad\\textrm{divide by term on RHS}\\\\ A_2 &= \\frac{A_1}{\\sqrt{2gh\\left(\\frac{A_1}{q}\\right)^2+1}}\\qquad\\textrm{take the square root}\\end{align}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\(A_2=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "sqrt((A_1^2*q^2)/(2*A_1^2*g*h+q^2))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a_1", "value": ""}, {"name": "g", "value": ""}, {"name": "h", "value": ""}, {"name": "q", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Engineering examples", "pickingStrategy": "random-subset", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "Orifice plate flowmeters", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "A difficult question that involves rearranging a complicated formula, then applying unit conversions to variable values, then evaluating the formula for the selected value. The variable values are randomised.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "In the design of orifice plate flowmeters, the volumetric flowrate, \\(Q\\) (m\\(^3\\)s\\(^{-1}\\)), is given by \\[Q=C_dA_o\\sqrt{\\frac{2gh}{1-A_o^2/A_p^2}}\\] where \\(C_d\\) is a dimensionless discharge coefficient, \\(h\\) (m) is the head difference across the orifice plate, \\(A_o\\) (m\\(^2\\)) is the area of the orifice and \\(A_p\\) (m\\(^2\\)) is the area of the pipe.
", "advice": "$\\begin{align}Q&=C_dA_o\\sqrt{\\frac{2gh}{1-A_o^2/A_p^2}}\\\\ \\frac{Q}{C_dA_o}&=\\sqrt{\\frac{2gh}{1-A_o^2/A_p^2}}\\\\ \\left(\\frac{Q}{C_dA_o}\\right)^2&=\\frac{2gh}{1-A_o^2/A_p^2}\\\\ \\left(\\frac{C_dA_o}{Q}\\right)^2&=\\frac{1-A_o^2/A_p^2}{2gh}\\\\ \\left(\\frac{C_d}{Q}\\right)^2A_o^2&=\\frac{1}{2gh}-\\frac{A_o^2}{2ghA_p^2}\\\\ \\left(\\frac{C_d}{Q}\\right)^2A_o^2 + \\frac{A_o^2}{2ghA_p^2} &=\\frac{1}{2gh}\\\\ \\left[\\frac{C_d^2}{Q^2} + \\frac{1}{2ghA_p^2}\\right]A_o^2 &=\\frac{1}{2gh}\\\\ \\left[\\frac{C_d^22ghA_p^2+Q^2}{Q^22ghA_p^2}\\right]A_o^2 &=\\frac{1}{2gh}\\\\ A_o^2&=\\frac{1}{2gh}\\times\\frac{Q^22ghA_p^2}{C_d^22ghA_p^2+Q^2}\\\\A_o&=\\frac{QA_p}{\\sqrt{C_d^22ghA_p^2+Q^2}}\\end{align}
\n$\\sqrt{\\var{Ao_eval_on_pi}}$
\nTo begin with, we need to work out the values of each of the variables in the correct units for the equation.
\n$Q = \\var{Q}$ cm$^3$s$^{-1}$ = $\\var{Q}\\div 1000000$ m$^3$s$^{-1}= \\var{Q_eval}$ m$^3$s$^{-1}$
\nPipe diameter = $\\var{A_p}$ cm = $\\var{A_p} \\div 100$ m = $\\var{2*Apr_eval}$ m
\nPipe radius = Pipe diameter $\\div 2 = \\var{2*Apr_eval}\\div 2 = \\var{Apr_eval}$ m
\nA$_p = \\pi\\times$ pipe radius$^2 = \\pi\\times \\var{Apr_eval}^2 = \\var{Ap_eval}$ m$^2$
\nC_d = $0.8$
\ng = $9.81$ ms$^{-2}$
\nh = $\\var{h} $ mm = $\\var{h}\\div 1000$ m = $\\var{h_eval}$ m
\nSo:
\n\\[ \\begin{align}A_o&=\\frac{QA_p}{\\sqrt{C_d^22ghA_p^2+Q^2}}\\\\
&= \\frac{\\var{Q_eval}\\times\\var{dpformat(Ap_eval,5,\"scientific\")}}{\\sqrt{(\\var{C_d})^2\\times 2\\times 9.81 \\times \\var{h_eval}\\times (\\var{dpformat(Ap_eval,5,\"scientific\")})^2+(\\var{Q_eval})^2}} \\\\
&= \\frac{\\var{numerator}}{\\sqrt{\\var{den_under_root_term1}+\\var{dpformat(Q_eval*Q_eval,5,\"scientific\")}}} \\\\
&= \\frac{\\var{numerator}}{\\sqrt{\\var{den_under_root}}}\\\\
&= \\frac{\\var{numerator}}{\\var{denominator}}\\\\
&=\\var{Ao_eval}\\text{ m}\\\\
\\text{And } A_o &= \\pi \\times \\text{ radius}^2\\\\
\\var{Ao_eval} & = \\pi\\times \\text{ radius}^2\\\\
\\text{radius}^2 & =\\frac{\\var{Ao_eval}}{\\pi} \\\\
& = \\var{Ao_eval_on_pi}\\\\
\\text{radius}&=\\sqrt{\\var{Ao_eval_on_pi}}\\\\
\\text{radius}&=\\var{root_Ao_eval_on_pi} \\text{ m}\\\\
& =\\var{root_Ao_eval_on_pi}\\times 100 \\text{ cm}\\\\
& =\\var{root_Ao_eval_on_pi*100} \\text{ cm}\\\\
\\text{Since diameter }&=\\text{ radius }\\times 2\\\\
\\text{diameter } &= \\var{root_Ao_eval_on_pi*100}\\times 2 \\text{ cm}\\\\
\\text{diameter } &= \\var{precround(root_Ao_eval_on_pi*100*2,3)} \\text{ cm}\\\\
\\end{align}\\]
used in the advice. The value inside the denominator square root for area Ao
", "templateType": "anything", "can_override": false}, "numerator": {"name": "numerator", "group": "Ungrouped variables", "definition": "dpformat(Q_eval*Ap_eval,5,\"scientific\")", "description": "used in the advice. The value of the numerator of the Area A_o
", "templateType": "anything", "can_override": false}, "denominator": {"name": "denominator", "group": "Ungrouped variables", "definition": "dpformat(sqrt(C_d^2*2*9.81*h_eval*Ap_eval^2+Q_eval^2),5,\"scientific\")", "description": "Used in the advice. The denominator of the area A_o
", "templateType": "anything", "can_override": false}, "test": {"name": "test", "group": "Ungrouped variables", "definition": "\"k\"", "description": "", "templateType": "anything", "can_override": false}, "den_under_root_term1": {"name": "den_under_root_term1", "group": "Ungrouped variables", "definition": "dpformat(C_d^2*2*9.81*h_eval*Ap_eval^2,5,\"scientific\")", "description": "used in the advice.
", "templateType": "anything", "can_override": false}, "Ao_eval": {"name": "Ao_eval", "group": "Ungrouped variables", "definition": "dpformat((Q_eval*Ap_eval)/(sqrt(C_d^2*2*9.81*h_eval*Ap_eval^2+Q_eval^2)),5,\"scientific\")", "description": "", "templateType": "anything", "can_override": false}, "Ao_eval_on_pi": {"name": "Ao_eval_on_pi", "group": "Ungrouped variables", "definition": "dpformat((Q_eval*Ap_eval)/(sqrt(C_d^2*2*9.81*h_eval*Ap_eval^2+Q_eval^2))/pi,5,\"scientific\")", "description": "used in the advice
", "templateType": "anything", "can_override": false}, "root_Ao_eval_on_pi": {"name": "root_Ao_eval_on_pi", "group": "Ungrouped variables", "definition": "siground(sqrt((Q_eval*Ap_eval)/(sqrt(C_d^2*2*9.81*h_eval*Ap_eval^2+Q_eval^2))/pi),5)", "description": "used in the advice
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Q", "A_p", "C_d", "h", "answer", "Q_eval", "Apr_eval", "Ap_eval", "h_eval", "den_under_root_term1", "den_under_root", "numerator", "denominator", "test", "Ao_eval", "Ao_eval_on_pi", "root_Ao_eval_on_pi"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Rearrange the equation to solve for the area of the orifice, \\(A_o\\), in terms of the other variables. (Note: to enter the variable name \\(A_1\\) type A_1, for \\(\\sqrt{a/b}\\) type sqrt(a/b).)
\\(A_o=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "Q*A_p/(sqrt(Q^2+2*g*h*A_p^2*C_d^2))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a_p", "value": ""}, {"name": "c_d", "value": ""}, {"name": "g", "value": ""}, {"name": "h", "value": ""}, {"name": "q", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "A volumetric flowrate of \\(\\var{Q}\\) cm\\(^3\\)s\\(^{-1}\\) passes through a \\(\\var{A_p}\\) cm inside diameter pipe. Assuming a discharge coefficient of \\(\\var{C_d}\\), calculate the required orifice diameter, so that the head difference across the orifice plate is \\(\\var{h}\\) mm. You may assume that \\(g=9.81\\) ms\\(^{-2}\\). Give your answer correct to 3 decimal places.
\n[Hint: be very careful with units.]
\norifice diameter\\(=\\)[[0]] cm
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{answer}", "maxValue": "{answer}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Coefficient of restitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Rearrange an equation for a variable e in k.1/(1-e) and then evaluate for e, given values for the variables.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "When a ball is dropped from rest onto a horizontal surface it will bounce before eventually coming to rest after a time \\(T\\) where \\[T=\\frac{2v}{g}\\left(\\frac{1}{1-e}\\right)\\] where \\(v\\) is the speed immediately after the first impact, and \\(g\\) is the acceleration due to gravity.
", "advice": "$\\begin{align}
T&=\\frac{2v}{g}\\left(\\frac{1}{1-e}\\right)\\\\
\\frac{Tg}{2v} &= \\left(\\frac{1}{1-e}\\right)\\\\
\\frac{2v}{Tg} &= \\left(\\frac{1-e}{1}\\right)\\\\
\\frac{2v}{Tg} &=1-e\\\\
e&=1-\\frac{2v}{Tg}
\\end{align}$
$\\begin{align}
e&=1-\\frac{2v}{Tg}\\\\
&=1-\\frac{2\\times\\var{velocity[velocity_s]}}{\\var{time}\\times 9.8}\\\\
&=1-\\var{(2*velocity[velocity_s])/(time*9.8)}\\\\
&\\approx\\var{precround(1-(2*velocity[velocity_s])/(time*9.8),2)}
\\end{align}$
Transpose the formula to make \\(e\\), the coefficient of restitution, the subject.
\n\\(e=\\)[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1-(2*v)/(T*g)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "g", "value": ""}, {"name": "t", "value": ""}, {"name": "v", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "A {balls[ball_s]} ball is dropped vertically from rest onto a horizontal {blocks[block_s]} surface. The ball's speed immediately after its first impact is \\(v = \\var{velocity[velocity_s]}\\) m/s and it comes to rest after \\(T=\\var{time}\\) s. Assuming that \\(g=9.8\\) m/s\\(^2\\), estimate the coefficient of restitution for this ball/surface combination.
\n\\(e=\\)[[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1-(2*velocity[velocity_s])/(time*9.8)", "maxValue": "1-(2*velocity[velocity_s])/(time*9.8)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "menu", "onleave": {"action": "none", "message": ""}, "preventleave": true, "typeendtoleave": false, "startpassword": "", "allowAttemptDownload": false, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "This quiz is a Numbas implementation of the Helping Engineers Learn Maths (HELM) booklet 1.5, Formulae and Transposition exercises.
\nQuestions generally have multiple versions, clicking the \"Try another question like this one\" button will generate a new version.
\nStarting a new instance of this quiz will create different questions in all sections.
", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Stephen Weissenhofer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2144/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "extensions": [], "custom_part_types": [], "resources": []}