// Numbas version: exam_results_page_options
{"name": "David's copy of Use formulae for the area and volume of geometric shapes", "extensions": ["geogebra"], "custom_part_types": [], "resources": [["question-resources/icecrea_QqVaCIf.svg", "/srv/numbas/media/question-resources/icecrea_QqVaCIf.svg"], ["question-resources/frisbee_variable_TESZa4J.svg", "/srv/numbas/media/question-resources/frisbee_variable_TESZa4J.svg"], ["question-resources/tennis-ball_with_variable_MBOLQeM.svg", "/srv/numbas/media/question-resources/tennis-ball_with_variable_MBOLQeM.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"name": "Trapezium variables", "variables": ["trap_a", "trap_b", "trap_c", "trap_d", "trap_h", "trap_rand", "trap_defs", "trap_length_a", "trap_length_b", "trap_e", "trap_areadp1", "trap_areadp2"]}, {"name": "Triangle variables", "variables": ["Triangle_area", "c", "b", "a", "c_theta", "length_a", "length_b", "length_c", "defs", "length_bdp2", "length_cdp2", "c_thetadp2"]}, {"name": "Name variables", "variables": ["name", "name2", "pronoun", "n"]}, {"name": "Quadratic variables", "variables": ["x2", "x1", "const"]}, {"name": "Cone variables", "variables": ["r", "h"]}, {"name": "RNG", "variables": ["mccall"]}], "ungrouped_variables": [], "variables": {"b": {"templateType": "anything", "group": "Triangle variables", "definition": "vector(-3,0)", "name": "b", "description": "<p>Position of the point A in Geogebra. This point is fixed so the triangle doesn't hang in one corner or the whole page.</p>"}, "length_b": {"templateType": "anything", "group": "Triangle variables", "definition": "sqrt((a[0]-c[0])^2+(a[1]-c[1])^2)", "name": "length_b", "description": "<p>For triangle - The length of the vector AC&nbsp;</p>"}, "trap_h": {"templateType": "anything", "group": "Trapezium variables", "definition": "trap_b[1] + 4", "name": "trap_h", "description": "<p>Height of the trapezium</p>"}, "c": {"templateType": "anything", "group": "Triangle variables", "definition": "vector(\n  random(2..5#0.1),\n  random(2..5#0.1)\n  )", "name": "c", "description": "<p>Triangle - A variable point which ultimately decides how the triangle looks.</p>"}, "trap_e": {"templateType": "anything", "group": "Trapezium variables", "definition": "vector(trap_b[0],-4)", "name": "trap_e", "description": "<p>Defines the point for the height of the trapezium.</p>"}, "c_thetadp2": {"templateType": "anything", "group": "Triangle variables", "definition": "precround((180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))),2)", "name": "c_thetadp2", "description": "<p>Rounded theta value.&nbsp;</p>"}, "x1": {"templateType": "anything", "group": "Quadratic variables", "definition": "random(1..50)", "name": "x1", "description": "<p>The x coefficient</p>"}, "mccall": {"templateType": "anything", "group": "RNG", "definition": "[0,random(3.1..3.7#0.1),random(5..20#0.1)]\n", "name": "mccall", "description": "<p>Matrix of random variables used to create length in the questions.</p>"}, "trap_d": {"templateType": "anything", "group": "Trapezium variables", "definition": "vector(random(5..7#0.1), -4)", "name": "trap_d", "description": "<p>Creates the point D on the trapezium</p>"}, "name2": {"templateType": "anything", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "name": "name2", "description": "<p>List of names to randomise. Can change to any name inserted</p>"}, "trap_length_b": {"templateType": "anything", "group": "Trapezium variables", "definition": "sqrt((trap_d[0]-trap_a[0])^2+(trap_d[1]-trap_a[1])^2)", "name": "trap_length_b", "description": ""}, "pronoun": {"templateType": "anything", "group": "Name variables", "definition": "if(mod(n,2)=0,\"he\",\"she\")", "name": "pronoun", "description": "<p>Defines the pronoun in the question.</p>"}, "c_theta": {"templateType": "anything", "group": "Triangle variables", "definition": "(180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))) ", "name": "c_theta", "description": "<p>Theta is randomised by the lengths&nbsp;</p>"}, "trap_c": {"templateType": "anything", "group": "Trapezium variables", "definition": "vector(random(4..5.5#0.1), trap_rand)", "name": "trap_c", "description": "<p>Creates the point C on the trapezium</p>"}, "trap_rand": {"templateType": "anything", "group": "Trapezium variables", "definition": "random(1..3#1)", "name": "trap_rand", "description": "<p>A random number to define the height of the trapezium.</p>"}, "x2": {"templateType": "anything", "group": "Quadratic variables", "definition": "random(1..10#1)", "name": "x2", "description": "<p>The x^2 coefficient</p>"}, "Triangle_area": {"templateType": "anything", "group": "Triangle variables", "definition": "1/2*(length_cdp2)(length_bdp2)(sin(c_thetadp2 * pi/180))", "name": "Triangle_area", "description": "<p>This calculates the area of the triangle for part&nbsp;<strong>b)</strong></p>"}, "length_bdp2": {"templateType": "anything", "group": "Triangle variables", "definition": "precround(sqrt((a[0]-c[0])^2+(a[1]-c[1])^2),2)", "name": "length_bdp2", "description": "<p>Rounded value for the length of b.</p>"}, "h": {"templateType": "anything", "group": "Cone variables", "definition": "random(11..17#0.1)", "name": "h", "description": "<p>The height&nbsp;for volume of a cone.</p>"}, "length_a": {"templateType": "anything", "group": "Triangle variables", "definition": "sqrt((a[0]-b[0])^2+(a[1]-b[1])^2)", "name": "length_a", "description": "<p>For triangle - The length of the vector AB</p>"}, "length_c": {"templateType": "anything", "group": "Triangle variables", "definition": "sqrt((b[0]-c[0])^2+(b[1]-c[1])^2)", "name": "length_c", "description": "<p>For triangle - The length of the vector BC</p>"}, "const": {"templateType": "anything", "group": "Quadratic variables", "definition": "random(1..100#1)", "name": "const", "description": "<p>The constant coefficient</p>"}, "n": {"templateType": "anything", "group": "Name variables", "definition": "random(0..4#1)", "name": "n", "description": "<p>n is a random number between 0 and 4 that picks a name from {name} and then picks the next in the list for the other name such that there is always a male and a female in the question.&nbsp;</p>"}, "trap_b": {"templateType": "anything", "group": "Trapezium variables", "definition": "vector(random(1.5..2.5#0.1), trap_rand)", "name": "trap_b", "description": "<p>Creates the point B on the trapezium</p>"}, "trap_length_a": {"templateType": "anything", "group": "Trapezium variables", "definition": "sqrt((trap_c[0]-trap_b[0])^2+(trap_c[1]-trap_b[1])^2)", "name": "trap_length_a", "description": ""}, "a": {"templateType": "anything", "group": "Triangle variables", "definition": "vector(2,0)", "name": "a", "description": "<p>Position of point B in Geogebra. This point is randomised to make the triangles different.</p>"}, "trap_defs": {"templateType": "anything", "group": "Trapezium variables", "definition": "[\n  ['A', trap_a],\n  ['B', trap_b],\n  ['C', trap_c],\n  ['D', trap_d],\n  ['E', trap_e]\n ]", "name": "trap_defs", "description": "<p>Definition of the points to put into Geogebra</p>"}, "r": {"templateType": "anything", "group": "Cone variables", "definition": "random(3..6#0.1)", "name": "r", "description": "<p>A random variable which will be inputted by the student.</p>"}, "trap_a": {"templateType": "anything", "group": "Trapezium variables", "definition": "vector(1,-4)", "name": "trap_a", "description": "<p>Creates the point A on the trapezium</p>"}, "trap_areadp2": {"templateType": "anything", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h, 2)", "name": "trap_areadp2", "description": ""}, "length_cdp2": {"templateType": "anything", "group": "Triangle variables", "definition": "precround(sqrt((b[0]-c[0])^2+(b[1]-c[1])^2),2)", "name": "length_cdp2", "description": "<p>Rounded value for the length of c.</p>"}, "defs": {"templateType": "anything", "group": "Triangle variables", "definition": "[\n  ['A',a],\n  ['B',b],\n  ['C',c]\n  ]", "name": "defs", "description": "<p>Creates the points in Geogebra is not used directly in the question but to create the image in Geogebra.</p>"}, "name": {"templateType": "anything", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "name": "name", "description": "<p>List of names to randomise. Can change to any name inserted</p>"}, "trap_areadp1": {"templateType": "anything", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h,1)", "name": "trap_areadp1", "description": "<p>Calculates the area of the trapezium</p>"}}, "metadata": {"description": "<p>Substitute values into&nbsp;formulae&nbsp;for the area or volume of various geometric objects.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Answer the following questions by substituting the correct&nbsp;values into the given equations.</p>", "preamble": {"js": "", "css": ""}, "rulesets": {}, "parts": [{"showCorrectAnswer": true, "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "minValue": "pi*{mccall[2]}^2 - 0.05", "allowFractions": false, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precision": "2", "strictPrecision": false, "variableReplacements": [], "precisionType": "dp", "maxValue": "pi*{mccall[2]}^2 + 0.05", "showCorrectAnswer": true, "showPrecisionHint": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "type": "numberentry", "correctAnswerStyle": "plain", "precisionPartialCredit": 0}], "scripts": {}, "prompt": "<p>Calculate the area of a frisbee, assuming that the frisbee can be modelled as a circle, given the formula for the area of a circle is</p>\n<p style=\"text-align: center;\">\\[\\mathrm{Area} = \\pi r^2.\\]</p>\n<p style=\"text-align: center;\"><object width=\"320\" height=\"320\" type=\"image/svg+xml\" data=\"resources/question-resources/frisbee_variable_TESZa4J.svg\"></object></p>\n<p>$\\mathrm{Area}$ = [[0]] $\\mathrm{cm}^2$ &nbsp; &nbsp;<em><small>Round your answer to 2 decimal places.</small></em></p>", "variableReplacements": []}, {"showCorrectAnswer": true, "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "minValue": "Triangle_area-0.05", "allowFractions": false, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precision": "2", "strictPrecision": false, "variableReplacements": [], "precisionType": "dp", "maxValue": "Triangle_area+0.05", "showCorrectAnswer": true, "showPrecisionHint": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "type": "numberentry", "correctAnswerStyle": "plain", "precisionPartialCredit": 0}], "scripts": {}, "prompt": "<p>Calculate the area of the triangle given that the area of any triangle can be calculated using the formula&nbsp;</p>\n<p style=\"text-align: center;\">\\[\\mathrm{Area} = \\frac{1}{2}ab\\sin{C}.\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('https://www.geogebra.org/m/jcUJu6F4',defs)}</p>\n<p style=\"text-align: left;\">All lengths are in centimetres.</p>\n<p>$\\mathrm{Area} =$&nbsp;[[0]] $\\mathrm{cm}^2$ &nbsp; <em><small>Round your answer to 2 decimal places.</small></em></p>", "variableReplacements": []}, {"showCorrectAnswer": true, "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "minValue": "{(h/3)}*{r}^2*pi -0.5", "allowFractions": false, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precision": "1", "strictPrecision": false, "variableReplacements": [], "precisionType": "dp", "maxValue": "{(h/3)}*{r}^2*pi +0.5", "showCorrectAnswer": true, "showPrecisionHint": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "type": "numberentry", "correctAnswerStyle": "plain", "precisionPartialCredit": 0}], "scripts": {}, "prompt": "<p>Calculate the volume of a cone given the formula for the volume of a cone is</p>\n<p style=\"text-align: center;\">\\[\\mathrm{Volume} = \\frac{h}{3} \\pi r^2.\\]</p>\n<p style=\"text-align: center;\"><object width=\"300\" height=\"250\" type=\"image/svg+xml\" data=\"resources/question-resources/icecrea_QqVaCIf.svg\"></object></p>\n<p style=\"text-align: left;\">$\\mathrm{Volume}$ =&nbsp;[[0]] $\\mathrm{cm}^3$ &nbsp; <em><small>Round your answer to 1 decimal place.</small></em></p>\n<p></p>", "variableReplacements": []}, {"showCorrectAnswer": true, "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "minValue": "4/3 * pi * {mccall[1]}^3 - 0.5", "allowFractions": false, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precision": "1", "strictPrecision": false, "variableReplacements": [], "precisionType": "dp", "maxValue": "4/3 * pi * {mccall[1]}^3 + 0.5", "showCorrectAnswer": true, "showPrecisionHint": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "type": "numberentry", "correctAnswerStyle": "plain", "precisionPartialCredit": 0}], "scripts": {}, "prompt": "<p>{name[n]} has a tennis ball and {pronoun} wants to find the volume of the ball. Using the diagram and the formula for the volume of a sphere, calculate the volume of the ball.&nbsp;</p>\n<p style=\"text-align: center;\">\\[\\mathrm{Volume}= \\frac{4}{3} \\pi r^3.\\]</p>\n<p style=\"text-align: center;\"><object width=\"220\" height=\"200\" type=\"image/svg+xml\" data=\"resources/question-resources/tennis-ball_with_variable_MBOLQeM.svg\"></object></p>\n<p>$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ &nbsp; <em><small>Round your answer to 1 decimal place.</small></em></p>", "variableReplacements": []}, {"showCorrectAnswer": true, "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "minValue": "0.5({trap_length_a}+{trap_length_b})*{trap_h} - 0.1", "allowFractions": false, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precision": "1", "strictPrecision": false, "variableReplacements": [], "precisionType": "dp", "maxValue": "0.5({trap_length_a}+{trap_length_b})*{trap_h} + 0.1", "showCorrectAnswer": true, "showPrecisionHint": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "type": "numberentry", "correctAnswerStyle": "plain", "precisionPartialCredit": 0}], "scripts": {}, "prompt": "<p>Find the area of the trapezium given the formula for the area of a trapezium is</p>\n<p style=\"text-align: center;\">\\[\\mathrm{Area} = \\frac{1}{2}(a+b) h .\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('https://www.geogebra.org/m/Gtjzajb6',trap_defs)}</p>\n<p><align></align></p>\n<p style=\"text-align: left;\">All&nbsp;lengths are given in metres.</p>\n<p>$\\mathrm{Area}$ = [[0]] $\\mathrm{m}^2$ &nbsp; <em><small>Round your answer to 1 decimal place.</small></em></p>\n<p></p>", "variableReplacements": []}], "functions": {}, "advice": "<p>When inserting numbers into your calculator, make sure that you place brackets correctly.</p>\n<h4>a)&nbsp;</h4>\n<p>We can see from the diagram that the radius of the frisbee is $\\var{mccall[2]}$ $\\mathrm{cm}$. <br/>Replacing the letter $r$ in the formula for the area of a circle with $\\var{mccall[2]}$ gives,</p>\n<p>\\begin{align}<br/>\\mathrm{Area} &amp;= \\pi r^2 \\\\<br/>&amp;= \\pi\\times(\\var{mccall[2]})^2 \\\\<br/>&amp;= \\var{dpformat((mccall[2])^2, 2)}\\pi\\, \\mathrm{cm}^2&nbsp;\\\\<br/>&amp;= \\var{dpformat(pi (mccall[2])^2, 2)}\\, \\mathrm{cm}^2 \\quad&nbsp;\\text{to 2 d.p.} &nbsp;<br/>\\end{align}</p>\n<h4>b)&nbsp;</h4>\n<p>We can see from the diagram that the triangle has two sides with lengths $\\var{length_cdp2}$ $\\mathrm{cm}$, $\\var{length_bdp2}$ $\\mathrm{cm}$ and an angle $\\var{c_thetadp2}\\mathrm{&deg;}$ .<br/>Replacing the letters $a$, $b$ and $C$ in the formula for the area of a triangle with $\\var{length_cdp2}$, $\\var{length_bdp2}$ and $\\var{c_thetadp2}$ respectively gives,</p>\n<p></p>\n<p>\\begin{align}<br/>\\mathrm{Area} &amp;= \\frac{1}{2}ab\\sin{C} \\\\<br/>&amp;= \\frac{1}{2} \\times \\var{length_cdp2} \\times \\var{length_bdp2} \\times \\sin(\\var{c_thetadp2}) \\\\<br/>&amp;= \\var{dpformat(0.5*(length_cdp2)*(length_bdp2)*sin(c_thetadp2* pi/180), 5)}\\, \\mathrm{cm}^2 &nbsp;\\\\<br/>&amp;= \\var{dpformat(0.5*(length_cdp2)*(length_bdp2)*sin(c_thetadp2 * pi/180), 2)}\\, \\mathrm{cm}^2 \\quad&nbsp;\\text{to 2 d.p.}<br/>\\end{align}</p>\n<p></p>\n<h4>c)&nbsp;</h4>\n<p>We can see from the diagram that the radius of the cone is $\\var{r}$ $\\mathrm{cm}$ and the height is $\\var{h}$ $\\mathrm{cm}$.<br/>Replacing the letters $r$ and $h$ in the formula for the volume of a cone with $\\var{r}$ $\\mathrm{cm}$ and $\\var{h}$ $\\mathrm{cm}$ respectively gives,</p>\n<p>\\begin{align}<br/>\\mathrm{Volume} &amp;= \\frac{h}{3} \\pi r^2 \\\\<br/>&amp;= \\frac{(\\var{h})}{3} \\times \\pi \\times (\\var{r})^2 \\\\<br/>&amp;= \\var{dpformat((pi)*(h/3)*(r)^2 , 5)}\\, \\mathrm{cm}^3 \\\\<br/>&amp;=\\var{dpformat(h/3 * pi * (r)^2, 1)}\\, \\mathrm{cm}^3 \\quad&nbsp;\\text{to 1 d.p.} \\\\<br/>\\end{align}</p>\n<p><em><small></small></em></p>\n<h4>d)</h4>\n<p>We can see from the diagram that the radius of the tennis ball is $\\var{mccall[1]}$ $\\mathrm{cm}$.<br/>Replacing the letter $r$ in the formula for the volume of a sphere with $\\var{mccall[1]}$ gives,</p>\n<p>\\begin{align}<br/>\\mathrm{Volume} &amp;= \\frac{4}{3} \\pi r^3 \\\\<br/>&amp;= \\frac{(4)}{(3)} \\times \\pi \\times (\\var{mccall[1]})^3 \\\\<br/>&amp;= \\var{dpformat((4/3)*pi*mccall[1]^3, 5)}\\,&nbsp; \\mathrm{cm}^3 \\\\<br/>&amp;= \\var{precround(((4/3)* pi) *(mccall[1])^3, 1)}\\, \\mathrm{cm}^3 \\quad&nbsp;\\text{to 1 d.p.} \\\\<br/>\\end{align}</p>\n<p></p>\n<h4>e)</h4>\n<p>We can see from the diagram that the trapezium has two parallel sides with length $\\var{trap_length_a}$ $\\mathrm{cm}$, $\\var{trap_length_b}$ $\\mathrm{cm}$ and height $\\var{trap_h}$ $\\mathrm{cm}$.<br/>Replacing the letters $a$, $b$ and $h$ in the formula for the area of a trapezium with&nbsp;$\\var{trap_length_a}$, $\\var{trap_length_b}$ and&nbsp;$\\var{trap_h}$ respectively gives,&nbsp;</p>\n<p></p>\n<p><span data-jme-visible=\"trap_areadp1 = trap_areadp2\">\\begin{align} <br/>\\mathrm{Area} &amp;= \\frac{1}{2} (a + b) h \\\\ <br/>&amp;= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\ <br/>&amp;= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad&nbsp;\\text{to 1 d.p.} \\\\ <br/>\\end{align}</span></p>\n<p><span data-jme-visible=\"trap_areadp1 &lt;&gt; trap_areadp2\">\\begin{align} <br/>\\mathrm{Area} &amp;= \\frac{1}{2} (a + b) h \\\\ <br/>&amp;= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\ <br/>&amp;= \\var{precround((0.5)(trap_length_a +trap_length_b) trap_h, 2)}\\, \\mathrm{cm}^2 \\\\ <br/>&amp;= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad&nbsp;\\text{to 1 d.p.} \\\\ <br/>\\end{align}</span></p>", "tags": ["Area", "area", "area of a circle", "area of a trapezium", "Area of a triangle", "area of a triangle", "circle", "Circle", "Cone", "cone", "geometry", "taxonomy", "trapezium", "triangle", "Triangle", "Volume", "volume", "volume of a cone", "volume of a sphere"], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "David's copy of Use formulae for the area and volume of geometric shapes", "extensions": ["geogebra"], "type": "question", "contributors": [{"name": "David Rickard", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/451/"}]}]}], "contributors": [{"name": "David Rickard", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/451/"}]}