Substitute values into formulae for the area or volume of various geometric objects.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Answer the following questions by substituting the correct values into the given equations.
", "advice": "When inserting numbers into your calculator, make sure that you place brackets correctly.
\nWe can see from the diagram that the radius of the frisbee is $\\var{mccall[2]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the area of a circle with $\\var{mccall[2]}$ gives,
\\begin{align}
\\mathrm{Area} &= \\pi r^2 \\\\
&= \\pi\\times(\\var{mccall[2]})^2 \\\\
&= \\var{dpformat((mccall[2])^2, 2)}\\pi\\, \\mathrm{cm}^2 \\\\
&= \\var{dpformat(pi *(mccall[2])^2, 2)}\\, \\mathrm{cm}^2 \\quad \\text{to 2 d.p.}
\\end{align}
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 $\\var{c_thetadp2}\\mathrm{°}$ .
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,
\\begin{align}
\\mathrm{Area} &= \\frac{1}{2}ab\\sin{C} \\\\
&= \\frac{1}{2} \\times \\var{length_cdp2} \\times \\var{length_bdp2} \\times \\sin(\\var{c_thetadp2}) \\\\
&= \\var{dpformat(0.5*(length_cdp2)*(length_bdp2)*sin(c_thetadp2* pi/180), 5)}\\, \\mathrm{cm}^2 \\\\
&= \\var{dpformat(0.5*(length_cdp2)*(length_bdp2)*sin(c_thetadp2 * pi/180), 2)}\\, \\mathrm{cm}^2 \\quad \\text{to 2 d.p.}
\\end{align}
We can see from the diagram that the radius of the cone is $\\var{r}$ $\\mathrm{cm}$ and the height is $\\var{h}$ $\\mathrm{cm}$.
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,
\\begin{align}
\\mathrm{Volume} &= \\frac{h}{3} \\pi r^2 \\\\
&= \\frac{(\\var{h})}{3} \\times \\pi \\times (\\var{r})^2 \\\\
&= \\var{dpformat((pi)*(h/3)*(r)^2 , 5)}\\, \\mathrm{cm}^3 \\\\
&=\\var{dpformat(h/3 * pi * (r)^2, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
\n
We can see from the diagram that the radius of the tennis ball is $\\var{mccall[1]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the volume of a sphere with $\\var{mccall[1]}$ gives,
\\begin{align}
\\mathrm{Volume} &= \\frac{4}{3} \\pi r^3 \\\\
&= \\frac{(4)}{(3)} \\times \\pi \\times (\\var{mccall[1]})^3 \\\\
&= \\var{dpformat((4/3)*pi*mccall[1]^3, 5)}\\, \\mathrm{cm}^3 \\\\
&= \\var{precround(((4/3)* pi) *(mccall[1])^3, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
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 $\\var{trap_h}$ $\\mathrm{cm}$.
Replacing the letters $a$, $b$ and $h$ in the formula for the area of a trapezium with $\\var{trap_length_a}$, $\\var{trap_length_b}$ and $\\var{trap_h}$ respectively gives,
\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5)(trap_length_a +trap_length_b) trap_h, 2)}\\, \\mathrm{cm}^2 \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
Rounded value for the length of c.
", "templateType": "anything", "can_override": false}, "trap_e": {"name": "trap_e", "group": "Trapezium variables", "definition": "vector(trap_b[0],-4)", "description": "Defines the point for the height of the trapezium.
", "templateType": "anything", "can_override": false}, "r": {"name": "r", "group": "Cone variables", "definition": "random(3..6#0.1)", "description": "A random variable which will be inputted by the student.
", "templateType": "anything", "can_override": false}, "const": {"name": "const", "group": "Quadratic variables", "definition": "random(1..100#1)", "description": "The constant coefficient
", "templateType": "anything", "can_override": false}, "name": {"name": "name", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "description": "List of names to randomise. Can change to any name inserted
", "templateType": "anything", "can_override": false}, "length_bdp2": {"name": "length_bdp2", "group": "Triangle variables", "definition": "precround(sqrt((a[0]-c[0])^2+(a[1]-c[1])^2),2)", "description": "Rounded value for the length of b.
", "templateType": "anything", "can_override": false}, "length_c": {"name": "length_c", "group": "Triangle variables", "definition": "sqrt((b[0]-c[0])^2+(b[1]-c[1])^2)", "description": "For triangle - The length of the vector BC
", "templateType": "anything", "can_override": false}, "trap_length_b": {"name": "trap_length_b", "group": "Trapezium variables", "definition": "sqrt((trap_d[0]-trap_a[0])^2+(trap_d[1]-trap_a[1])^2)", "description": "", "templateType": "anything", "can_override": false}, "c_thetadp2": {"name": "c_thetadp2", "group": "Triangle variables", "definition": "precround((180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))),2)", "description": "Rounded theta value.
", "templateType": "anything", "can_override": false}, "trap_length_a": {"name": "trap_length_a", "group": "Trapezium variables", "definition": "sqrt((trap_c[0]-trap_b[0])^2+(trap_c[1]-trap_b[1])^2)", "description": "", "templateType": "anything", "can_override": false}, "length_b": {"name": "length_b", "group": "Triangle variables", "definition": "sqrt((a[0]-c[0])^2+(a[1]-c[1])^2)", "description": "For triangle - The length of the vector AC
", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Triangle variables", "definition": "vector(2,0)", "description": "Position of point B in Geogebra. This point is randomised to make the triangles different.
", "templateType": "anything", "can_override": false}, "pronoun": {"name": "pronoun", "group": "Name variables", "definition": "if(mod(n,2)=0,\"he\",\"she\")", "description": "Defines the pronoun in the question.
", "templateType": "anything", "can_override": false}, "trap_defs": {"name": "trap_defs", "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 ]", "description": "Definition of the points to put into Geogebra
", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Cone variables", "definition": "random(11..17#0.1)", "description": "The height for volume of a cone.
", "templateType": "anything", "can_override": false}, "trap_h": {"name": "trap_h", "group": "Trapezium variables", "definition": "trap_b[1] + 4", "description": "Height of the trapezium
", "templateType": "anything", "can_override": false}, "Triangle_area": {"name": "Triangle_area", "group": "Triangle variables", "definition": "1/2*(length_cdp2)(length_bdp2)(sin(c_thetadp2 * pi/180))", "description": "This calculates the area of the triangle for part b)
", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Triangle variables", "definition": "vector(-3,0)", "description": "Position of the point A in Geogebra. This point is fixed so the triangle doesn't hang in one corner or the whole page.
", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Name variables", "definition": "random(0..4#1)", "description": "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.
", "templateType": "anything", "can_override": false}, "name2": {"name": "name2", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "description": "List of names to randomise. Can change to any name inserted
", "templateType": "anything", "can_override": false}, "trap_rand": {"name": "trap_rand", "group": "Trapezium variables", "definition": "random(1..3#1)", "description": "A random number to define the height of the trapezium.
", "templateType": "anything", "can_override": false}, "c_theta": {"name": "c_theta", "group": "Triangle variables", "definition": "(180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))) ", "description": "Theta is randomised by the lengths
", "templateType": "anything", "can_override": false}, "trap_areadp1": {"name": "trap_areadp1", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h,1)", "description": "Calculates the area of the trapezium
", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Triangle variables", "definition": "[\n ['A',a],\n ['B',b],\n ['C',c]\n ]", "description": "Creates the points in Geogebra is not used directly in the question but to create the image in Geogebra.
", "templateType": "anything", "can_override": false}, "length_a": {"name": "length_a", "group": "Triangle variables", "definition": "sqrt((a[0]-b[0])^2+(a[1]-b[1])^2)", "description": "For triangle - The length of the vector AB
", "templateType": "anything", "can_override": false}, "trap_d": {"name": "trap_d", "group": "Trapezium variables", "definition": "vector(random(5..7#0.1), -4)", "description": "Creates the point D on the trapezium
", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Triangle variables", "definition": "vector(\n random(2..5#0.1),\n random(2..5#0.1)\n )", "description": "Triangle - A variable point which ultimately decides how the triangle looks.
", "templateType": "anything", "can_override": false}, "trap_a": {"name": "trap_a", "group": "Trapezium variables", "definition": "vector(1,-4)", "description": "Creates the point A on the trapezium
", "templateType": "anything", "can_override": false}, "trap_c": {"name": "trap_c", "group": "Trapezium variables", "definition": "vector(random(4..5.5#0.1), trap_rand)", "description": "Creates the point C on the trapezium
", "templateType": "anything", "can_override": false}, "mccall": {"name": "mccall", "group": "RNG", "definition": "[0,random(3.1..3.7#0.1),random(5..20#0.1)]\n", "description": "Matrix of random variables used to create length in the questions.
", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Quadratic variables", "definition": "random(1..10#1)", "description": "The x^2 coefficient
", "templateType": "anything", "can_override": false}, "trap_areadp2": {"name": "trap_areadp2", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h, 2)", "description": "", "templateType": "anything", "can_override": false}, "trap_b": {"name": "trap_b", "group": "Trapezium variables", "definition": "vector(random(1.5..2.5#0.1), trap_rand)", "description": "Creates the point B on the trapezium
", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Quadratic variables", "definition": "random(1..50)", "description": "The x coefficient
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "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"]}], "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": "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
\n\\[\\mathrm{Area} = \\pi r^2.\\]
\n\n$\\mathrm{Area}$ = [[0]] $\\mathrm{cm}^2$ Round your answer to 2 decimal places.
", "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": "pi*{mccall[2]}^2 - 0.05", "maxValue": "pi*{mccall[2]}^2 + 0.05", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "Calculate the area of the triangle given that the area of any triangle can be calculated using the formula
\n\\[\\mathrm{Area} = \\frac{1}{2}ab\\sin{C}.\\]
\n{geogebra_applet('https://www.geogebra.org/m/jcUJu6F4',defs)}
\nAll lengths are in centimetres.
\n$\\mathrm{Area} =$ [[0]] $\\mathrm{cm}^2$ Round your answer to 2 decimal places.
", "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": "Triangle_area-0.05", "maxValue": "Triangle_area+0.05", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "Calculate the volume of a cone given the formula for the volume of a cone is
\n\\[\\mathrm{Volume} = \\frac{h}{3} \\pi r^2.\\]
\n\n$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.
\n", "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": "{(h/3)}*{r}^2*pi -0.5", "maxValue": "{(h/3)}*{r}^2*pi +0.5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "{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.
\n\\[\\mathrm{Volume}= \\frac{4}{3} \\pi r^3.\\]
\n\n$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.
", "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": "4/3 * pi * {mccall[1]}^3 - 0.5", "maxValue": "4/3 * pi * {mccall[1]}^3 + 0.5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "Find the area of the trapezium given the formula for the area of a trapezium is
\n\\[\\mathrm{Area} = \\frac{1}{2}(a+b) h .\\]
\n{geogebra_applet('https://www.geogebra.org/m/Gtjzajb6',trap_defs)}
\nAll lengths are given in metres.
\n$\\mathrm{Area}$ = [[0]] $\\mathrm{m}^2$ Round your answer to 1 decimal place.
\n", "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": "0.5({trap_length_a}+{trap_length_b})*{trap_h} - 0.1", "maxValue": "0.5({trap_length_a}+{trap_length_b})*{trap_h} + 0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "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", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}]}