// Numbas version: exam_results_page_options {"percentPass": 0, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "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}, "contributors": [{"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "rulesets": {}, "functions": {}, "ungrouped_variables": [], "metadata": {"description": "

Substitute values into formulae for the area or volume of various geometric objects.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [{"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": "Trapezium variables"}, {"variables": ["Triangle_area", "c", "b", "a", "c_theta", "length_a", "length_b", "length_c", "defs", "length_bdp2", "length_cdp2", "c_thetadp2"], "name": "Triangle variables"}, {"variables": ["name", "name2", "pronoun", "n"], "name": "Name variables"}, {"variables": ["x2", "x1", "const"], "name": "Quadratic variables"}, {"variables": ["r", "h"], "name": "Cone variables"}, {"variables": ["mccall"], "name": "RNG"}], "advice": "

When inserting numbers into your calculator, make sure that you place brackets correctly.

\n

a)

\n

We 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,

\n

\\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}

\n

b)

\n

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,

\n

\n

\\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  \\\\
\\end{align}

\n

\n

c)

\n

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,

\n

\\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

\n

d)

\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,

\n

\\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}

\n

\n

e)

\n

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,

\n

\n

\\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}

\n

\\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}

", "statement": "

Answer the following questions by substituting the correct values into the given equations.

", "preamble": {"css": "", "js": ""}, "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"], "parts": [{"variableReplacementStrategy": "originalfirst", "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": [{"variableReplacementStrategy": "originalfirst", "maxValue": "pi*{mccall[2]}^2 + 0.05", "precision": "2", "scripts": {}, "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "precisionType": "dp", "allowFractions": false, "showPrecisionHint": false, "minValue": "pi*{mccall[2]}^2 - 0.05", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true}], "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "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)}

\n

All lengths are in centimetres.

\n

$\\mathrm{Area} =$ [[0]] $\\mathrm{cm}^2$   Round your answer to 2 decimal places.

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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

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{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": [{"variableReplacementStrategy": "originalfirst", "maxValue": "4/3 * pi * {mccall[1]}^3 + 0.5", "precision": "1", "scripts": {}, "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "precisionType": "dp", "allowFractions": false, "showPrecisionHint": false, "minValue": "4/3 * pi * {mccall[1]}^3 - 0.5", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true}], "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "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)}

\n

\n

All lengths are given in metres.

\n

$\\mathrm{Area}$ = [[0]] $\\mathrm{m}^2$   Round your answer to 1 decimal place.

\n

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Rounded value for the length of c.

", "name": "length_cdp2", "group": "Triangle variables", "definition": "precround(sqrt((b[0]-c[0])^2+(b[1]-c[1])^2),2)"}, "trap_e": {"templateType": "anything", "description": "

Defines the point for the height of the trapezium.

", "name": "trap_e", "group": "Trapezium variables", "definition": "vector(trap_b[0],-4)"}, "r": {"templateType": "anything", "description": "

A random variable which will be inputted by the student.

", "name": "r", "group": "Cone variables", "definition": "random(3..6#0.1)"}, "const": {"templateType": "anything", "description": "

The constant coefficient

", "name": "const", "group": "Quadratic variables", "definition": "random(1..100#1)"}, "name": {"templateType": "anything", "description": "

List of names to randomise. Can change to any name inserted

", "name": "name", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]"}, "length_bdp2": {"templateType": "anything", "description": "

Rounded value for the length of b.

", "name": "length_bdp2", "group": "Triangle variables", "definition": "precround(sqrt((a[0]-c[0])^2+(a[1]-c[1])^2),2)"}, "length_c": {"templateType": "anything", "description": "

For triangle - The length of the vector BC

", "name": "length_c", "group": "Triangle variables", "definition": "sqrt((b[0]-c[0])^2+(b[1]-c[1])^2)"}, "trap_length_b": {"templateType": "anything", "description": "", "name": "trap_length_b", "group": "Trapezium variables", "definition": "sqrt((trap_d[0]-trap_a[0])^2+(trap_d[1]-trap_a[1])^2)"}, "c_thetadp2": {"templateType": "anything", "description": "

Rounded theta value.

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For triangle - The length of the vector AC

", "name": "length_b", "group": "Triangle variables", "definition": "sqrt((a[0]-c[0])^2+(a[1]-c[1])^2)"}, "a": {"templateType": "anything", "description": "

Position of point B in Geogebra. This point is randomised to make the triangles different.

", "name": "a", "group": "Triangle variables", "definition": "vector(2,0)"}, "pronoun": {"templateType": "anything", "description": "

Defines the pronoun in the question.

", "name": "pronoun", "group": "Name variables", "definition": "if(mod(n,2)=0,\"he\",\"she\")"}, "trap_defs": {"templateType": "anything", "description": "

Definition of the points to put into Geogebra

", "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 ]"}, "h": {"templateType": "anything", "description": "

The height for volume of a cone.

", "name": "h", "group": "Cone variables", "definition": "random(11..17#0.1)"}, "trap_h": {"templateType": "anything", "description": "

Height of the trapezium

", "name": "trap_h", "group": "Trapezium variables", "definition": "trap_b[1] + 4"}, "Triangle_area": {"templateType": "anything", "description": "

This calculates the area of the triangle for part b)

", "name": "Triangle_area", "group": "Triangle variables", "definition": "1/2*(length_cdp2)(length_bdp2)(sin(c_thetadp2 * pi/180))"}, "b": {"templateType": "anything", "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.

", "name": "b", "group": "Triangle variables", "definition": "vector(-3,0)"}, "n": {"templateType": "anything", "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.

", "name": "n", "group": "Name variables", "definition": "random(0..4#1)"}, "name2": {"templateType": "anything", "description": "

List of names to randomise. Can change to any name inserted

", "name": "name2", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]"}, "trap_rand": {"templateType": "anything", "description": "

A random number to define the height of the trapezium.

", "name": "trap_rand", "group": "Trapezium variables", "definition": "random(1..3#1)"}, "c_theta": {"templateType": "anything", "description": "

Theta is randomised by the lengths

", "name": "c_theta", "group": "Triangle variables", "definition": "(180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))) "}, "trap_areadp1": {"templateType": "anything", "description": "

Calculates the area of the trapezium

", "name": "trap_areadp1", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h,1)"}, "defs": {"templateType": "anything", "description": "

Creates the points in Geogebra is not used directly in the question but to create the image in Geogebra.

", "name": "defs", "group": "Triangle variables", "definition": "[\n ['A',a],\n ['B',b],\n ['C',c]\n ]"}, "length_a": {"templateType": "anything", "description": "

For triangle - The length of the vector AB

", "name": "length_a", "group": "Triangle variables", "definition": "sqrt((a[0]-b[0])^2+(a[1]-b[1])^2)"}, "trap_d": {"templateType": "anything", "description": "

Creates the point D on the trapezium

", "name": "trap_d", "group": "Trapezium variables", "definition": "vector(random(5..7#0.1), -4)"}, "c": {"templateType": "anything", "description": "

Triangle - A variable point which ultimately decides how the triangle looks.

", "name": "c", "group": "Triangle variables", "definition": "vector(\n random(2..5#0.1),\n random(2..5#0.1)\n )"}, "trap_a": {"templateType": "anything", "description": "

Creates the point A on the trapezium

", "name": "trap_a", "group": "Trapezium variables", "definition": "vector(1,-4)"}, "trap_c": {"templateType": "anything", "description": "

Creates the point C on the trapezium

", "name": "trap_c", "group": "Trapezium variables", "definition": "vector(random(4..5.5#0.1), trap_rand)"}, "mccall": {"templateType": "anything", "description": "

Matrix of random variables used to create length in the questions.

", "name": "mccall", "group": "RNG", "definition": "[0,random(3.1..3.7#0.1),random(5..20#0.1)]\n"}, "x2": {"templateType": "anything", "description": "

The x^2 coefficient

", "name": "x2", "group": "Quadratic variables", "definition": "random(1..10#1)"}, "trap_areadp2": {"templateType": "anything", "description": "", "name": "trap_areadp2", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h, 2)"}, "trap_b": {"templateType": "anything", "description": "

Creates the point B on the trapezium

", "name": "trap_b", "group": "Trapezium variables", "definition": "vector(random(1.5..2.5#0.1), trap_rand)"}, "x1": {"templateType": "anything", "description": "

The x coefficient

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{person['name']} just bought a new house. {capitalise(pronouns['their'])} new bedroom's wall and ceiling are currently painted white, but {pronouns['they']} would like to paint these {colour}.

\n

The dimensions of the floor are $\\var{length}\\,\\mathrm{m} \\times \\var{width}\\,\\mathrm{m}$ and the room is $\\var{height}\\,\\mathrm{m}$ high.

\n

{person['name']} want{verbs} to know how much paint to buy so {pronouns['they']} can paint all four walls and the ceiling {colour}.

", "variablesTest": {"condition": "mod(rall,15)<>0", "maxRuns": 100}, "variables": {"buckets": {"description": "", "name": "buckets", "group": "Calculations", "templateType": "anything", "definition": "ceil(rall/bucket_area)"}, "bucket_area": {"description": "

The area that can be painted with one bucket of paint.

", "name": "bucket_area", "group": "Calculations", "templateType": "anything", "definition": "15"}, "colour": {"description": "", "name": "colour", "group": "Random bits", "templateType": "anything", "definition": "random(\"green\", \"red\", \"orange\", \"yellow\", \"blue\", \"purple\", \"pink\")"}, "width": {"description": "", "name": "width", "group": "Random bits", "templateType": "anything", "definition": "random(2.50..5.00 #0.1) + random(0.01..0.09 #0.01)"}, "w": {"description": "", "name": "w", "group": "Calculations", "templateType": "anything", "definition": "ceil(width)"}, "verbs": {"description": "", "name": "verbs", "group": "Person", "templateType": "anything", "definition": "if(person['gender']='neutral','','s')"}, "height": {"description": "", "name": "height", "group": "Random bits", "templateType": "anything", "definition": "random(2.10..2.70 #0.1) + random(0.01..0.09 #0.01)"}, "h": {"description": "", "name": "h", "group": "Calculations", "templateType": "anything", "definition": "ceil(height)"}, "rall": {"description": "", "name": "rall", "group": "Calculations", "templateType": "anything", "definition": "rceiling + rwall1*2 + rwall2*2"}, "pronouns": {"description": "", "name": "pronouns", "group": "Person", "templateType": "anything", "definition": "person['pronouns']"}, "length": {"description": "", "name": "length", "group": "Random bits", "templateType": "anything", "definition": "random(3.50..8.00 #0.1) + random(0.01..0.09 #0.01)"}, "person": {"description": "", "name": "person", "group": "Person", "templateType": "anything", "definition": "random_person()"}, "rwall1": {"description": "", "name": "rwall1", "group": "Calculations", "templateType": "anything", "definition": "l*h"}, "rwall2": {"description": "", "name": "rwall2", "group": "Calculations", "templateType": "anything", "definition": "w*h"}, "rceiling": {"description": "", "name": "rceiling", "group": "Calculations", "templateType": "anything", "definition": "w*l"}, "l": {"description": "", "name": "l", "group": "Calculations", "templateType": "anything", "definition": "ceil(length)"}}, "functions": {}, "tags": ["random names", "taxonomy"], "variable_groups": [{"name": "Random bits", "variables": ["colour", "height", "width", "length"]}, {"name": "Person", "variables": ["person", "pronouns", "verbs"]}, {"name": "Calculations", "variables": ["h", "w", "l", "rwall1", "rwall2", "rceiling", "rall", "bucket_area", "buckets"]}], "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", ""], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "choices": ["

Overestimate and therefore we round each measurement up.

", "

Underestimate and therefore we round each measurement down.

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Is it better to overestimate or underestimate in this situation?

\n

[[0]]

", "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "rall", "showFeedbackIcon": true, "minValue": "rall", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "stepsPenalty": 0, "prompt": "

Rounding each measurement to the nearest metre, estimate the whole area to be painted {colour}.

\n

[[0]] m2

\n

", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "

The room is {length}m long, {width}m wide, and {height}m high.

\n

Round each measurement in the direction you decided on above.

", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ceil(length)", "showFeedbackIcon": true, "prompt": "

Round the length to the nearest metre.

", "minValue": "ceil(length)", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": "0.1", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ceil(width)", "showFeedbackIcon": true, "prompt": "

Round the width to the nearest metre.

", "minValue": "ceil(width)", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": "0.1", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ceil(height)", "showFeedbackIcon": true, "prompt": "

Round the height to the nearest metre.

", "minValue": "ceil(height)", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": "0.1", "showCorrectAnswer": true}], "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "buckets", "showFeedbackIcon": true, "minValue": "buckets", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "

One bucket of {colour} paint is enough to paint an area of 15m2. How many buckets should {person['name']} buy to ensure {pronouns['they']} {if(person['gender']='neutral','have','has')} enough paint?

\n

[[0]]

", "marks": 0}], "ungrouped_variables": [], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Estimate the number of buckets of paint to buy, by rounding measurements of a room up to the nearest metre and estimating the total area.

"}, "preamble": {"css": "", "js": ""}, "advice": "

a)

\n

It is much better to have spare paint than not to have enough of it. So it is better to overestimate the area.

\n

Therefore, we round each measurement up.

\n

b)

\n

We round each of our measurements up to the nearest whole metre:

\n

Length: $\\var{length}\\,\\mathrm{m} \\approx \\var{l}\\,\\mathrm{m}$.

\n

Width: $\\var{width}\\,\\mathrm{m} \\approx \\var{w}\\,\\mathrm{m}$.

\n

Height: $\\var{height}\\,\\mathrm{m} \\approx \\var{h}\\,\\mathrm{m}$.

\n

The total area consists of five areas: two walls of $\\var{l}\\,\\text{m} \\times \\var{h}\\,\\text{m}$ (length by height); two walls of $\\var{w}\\,\\text{m} \\times \\var{h}\\,\\text{m}$ (width by height); and a ceiling of $\\var{l}\\,\\text{m} \\times \\var{w}\\,\\text{m}$ (length by width).

\n

\\\begin{align} \\var{l}\\,\\text{m} \\times \\var{h}\\,\\text{m} &= \\var{l*h}\\,\\text{m}^2 \\\\ \\var{w}\\,\\text{m} \\times \\var{h}\\,\\text{m} &= \\var{w*h}\\,\\text{m}^2 \\\\ \\var{l}\\,\\text{m} \\times \\var{w}\\,\\text{m} &= \\var{l*w}\\,\\text{m}^2 \\end{align}\

\n

Therefore, the total area {person['name']} needs to paint is

\n

\$\\var{2*l*h} + \\var{2*w*h} + \\var{l*w} \\,\\mathrm{m}^2 = \\var{rall}\\,\\mathrm{m}^2 \\text{.} \$

\n

c)

\n

The exact number of buckets needed is

\n

\$\\var{rall}\\,\\text{m}^2 \\div 15\\,\\text{m}^2 = \\var{rall/15} \\text{.}\$

\n

{person['name']} can only buy a whole number of buckets, so {pronouns['they']} need{verbs} to decide between {buckets-1} and {buckets} paint buckets. As it is better to buy more paint than not buy enough, {pronouns['they']} should buy {buckets} buckets of {colour} paint.

"}, {"name": "Calculate the areas of polygons", "extensions": [], "custom_part_types": [], "resources": [["question-resources/trapezium.svg", "/srv/numbas/media/question-resources/trapezium.svg"], ["question-resources/trangle.svg", "/srv/numbas/media/question-resources/trangle.svg"], ["question-resources/parallelogram.svg", "/srv/numbas/media/question-resources/parallelogram.svg"], ["question-resources/Parallelogram_area_animated.gif", "/srv/numbas/media/question-resources/Parallelogram_area_animated.gif"], ["question-resources/rectangle_zISmvoz.svg", "/srv/numbas/media/question-resources/rectangle_zISmvoz.svg"], ["question-resources/hardertrapezium_8GqMwOo.svg", "/srv/numbas/media/question-resources/hardertrapezium_8GqMwOo.svg"], ["question-resources/Trap_advice.svg", "/srv/numbas/media/question-resources/Trap_advice.svg"], ["question-resources/Triangle_advice_lD6eKvD.svg", "/srv/numbas/media/question-resources/Triangle_advice_lD6eKvD.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "rulesets": {}, "functions": {}, "ungrouped_variables": [], "metadata": {"description": "

This question tests the students ability to calculate the area of different 2D shapes given the units and measurements required. The formulae for the areas are available if required but students are encouraged to try to remember them themselves.

\n

The shapes are: a rectangle, a parallelogram, a right-angled triangle, and a trapezium.

\n

Author of gif: Picknick
https://commons.wikimedia.org/wiki/File:Parallelogram_area_animated.gif

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [{"variables": ["h1", "w1", "wh11", "wh11dp"], "name": "Parallelogram"}, {"variables": ["h2", "w2", "wh22", "wh22dp"], "name": "Triangle"}, {"variables": ["h5", "w5a", "w5b", "wabh5dp", "wabh5"], "name": "'Harder' trapezium"}, {"variables": ["w0", "h0", "wh00", "wh00dp"], "name": "Rectangle"}], "advice": "

a)

\n

The area of a rectangle is calculated using the formula

\n

\$\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}\\text{.}\$

\n

We have a base of $\\var{w0}$m and a height $\\var{h0}$m, therefore

\n

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w0} \\times \\var{h0} \\\\ &= \\var{w0*h0} \\\\
&= \\var{dpformat(w0*h0,1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w0} \\times \\var{h0} \\\\
&= \\var{dpformat(w0*h0,1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

b)

\n

The parallelogram is just a slanted rectangle:

\n

\n

\n
Animation by Picknick.
\n

\n

Therefore, the area of a parallelogram is calculated using the formula

\n

\$\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\$

\n

We have a base $\\var{w1}$m and perpendicular height $\\var{h1}$m.

\n

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w1} \\times \\var{h1} \\\\ &= \\var{{w1}{h1}}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat({w1}{h1},1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w1} \\times \\var{h1} \\\\
&= \\var{dpformat({w1}{h1},1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

c)

\n

The area of a triangle is calculated using the formula

\n

\$\\mathrm{Area} = \\frac{\\mathrm{base} \\times \\mathrm{height}}{2}.\$

\n

Note that the triangle is half of a rectangle:

\n

\n

Our triangle has a base $\\var{w2}$m and a height $\\var{h2}$m, therefore

\n

\\begin{align} \\mathrm{Area} &= \\frac{1}{2} \\times \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\frac{1}{2} \\times \\var{w2} \\times \\var{h2} \\\\
&= \\var{0.5*w2*h2}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat(0.5*w2*h2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

\\begin{align} \\mathrm{Area} &= \\frac{1}{2} \\times \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\frac{1}{2} \\times \\var{w2} \\times \\var{h2} \\\\
&= \\var{dpformat(0.5*w2*h2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\n

d)

\n

\n

A trapezium can be interpreted as half of a parallelogram, this is shown below:

\n

\n

As we only want the area of one half of this shape, the area is half of

\n

\$\\mathrm{area} = (a+b) \\times \\mathrm{height}\\text{,}\$

\n

with ${a} = \\var{w5a}$m, ${b} = \\var{w5b}$m, and height $\\var{h5}$m.

\n

\\begin{align}
\\mathrm{Area} &= \\frac{(a+b)}{2} \\times \\mathrm{height} \\\\
&= \\frac{(\\var{w5a}+\\var{w5b})}{2} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*0.5} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*(h5)/2}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat((w5a+w5b)*(h5)/2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.}
\\end{align}

\n

\\begin{align}
\\mathrm{Area} &= \\frac{(a+b)}{2} \\times \\mathrm{height} \\\\
&= \\frac{(\\var{w5a}+\\var{w5b})}{2} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*0.5} \\times \\var{h5} \\\\
&= \\var{dpformat((w5a+w5b)*(h5)/2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.}
\\end{align}

", "statement": "

Calculate the area of the following shapes.

", "preamble": {"css": "", "js": ""}, "tags": ["area", "Area", "area of a parallelogram", "area of a rectangle", "area of a right-angled triangle", "area of a trapezium", "parallelogram", "Rectangle", "rectangle", "right - angled triangle", "shapes", "taxonomy", "trapezium"], "parts": [{"variableReplacementStrategy": "originalfirst", "prompt": "

\n

The area of the rectangle is [[0]] $\\mathrm{m^2}$.      Round your answer to 1 decimal place.

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The formula for the area of a rectangle is:

\n

\$\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\$

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "prompt": "

\n

The area of the parallelogram is [[0]] $\\mathrm{m^2}$.       Round your answer to 1 decimal place.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "variableReplacements": [], "precision": "1", "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "maxValue": "{h1}*{w1} + 0.01", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "minValue": "{h1}*{w1} - 0.01", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "marks": "2", "scripts": {}}], "stepsPenalty": "1", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The formula for the area of a parallelogram is:

\n

\$\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\$

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "prompt": "

\n

The area of the triangle is [[0]] $\\mathrm{m^2}$      Round your answer to 1 decimal place.

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The formula for the area of a triangle is:

\n

\$\\mathrm{Area} = \\frac{\\mathrm{base} \\times \\mathrm{height}}{2}.\$

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "prompt": "

\n

The area of the trapezium is [[0]] $\\mathrm{m^2}$.    Round your answer to 1 decimal place.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "variableReplacements": [], "precision": "1", "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "maxValue": "0.5{w5a+w5b}{h5} + 0.01", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "minValue": "0.5{w5a+w5b}{h5} - 0.01", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "marks": "2", "scripts": {}}], "stepsPenalty": "1", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The formula for the area of a trapezium is:

\n

\$\\mathrm{Area} = \\frac{(a+b)}{2}\\times \\mathrm{height}.\$

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}], "variables": {"h2": {"name": "h2", "description": "

Height of the triangle.

", "templateType": "anything", "group": "Triangle", "definition": "random(1..4.5#0.1)"}, "wh22dp": {"name": "wh22dp", "description": "

The Area of a triangle using the two terms, w2 and h2 to one decimal place, such that a condition can be satisfied.

", "templateType": "anything", "group": "Triangle", "definition": "precround(0.5*w2*h2, 1)"}, "wabh5": {"name": "wabh5", "description": "

The Area of a trapezium using the three terms, w5a, w5b and h5, such that a condition can be satisfied.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "precround((w5a+w5b)*(h5)/2, 5)"}, "w5b": {"name": "w5b", "description": "

The bottom parallel side in the trapezium.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "random(7.5..10#0.1)"}, "wh11": {"name": "wh11", "description": "

The product of the two terms, w1 and h1, such that a condition can be satisfied.

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The width of the parallelogram.

", "templateType": "anything", "group": "Parallelogram", "definition": "random(5..10#0.1)"}, "wabh5dp": {"name": "wabh5dp", "description": "

The Area of a trapezium using the three terms, w5a, w5b and h5 to one decimal place, such that a condition can be satisfied.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "precround((w5a+w5b)*(h5)/2, 1)"}, "h1": {"name": "h1", "description": "

The height of the parallelogram

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Width of the rectangle.

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The Area of a triangle using the two terms, w2 and h2, such that a condition can be satisfied.

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Base of the triangle.

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Height of the trapezium.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "random(2..5#0.1)"}, "wh00dp": {"name": "wh00dp", "description": "

The product of the two terms, w0 and h0, to one decimal place, such that a condition can be satisfied.

", "templateType": "anything", "group": "Rectangle", "definition": "precround(w0*h0,1)"}, "wh00": {"name": "wh00", "description": "

The product of the two terms, w0 and h0, such that a condition can be satisfied.

", "templateType": "anything", "group": "Rectangle", "definition": "precround(w0*h0,3)"}, "h0": {"name": "h0", "description": "

Height of the rectangle.

", "templateType": "anything", "group": "Rectangle", "definition": "random(1..5#0.1)"}, "w5a": {"name": "w5a", "description": "

The top parallel side in the trapezium.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "random(5..6.5#0.1)"}, "wh11dp": {"name": "wh11dp", "description": "

The product of the two terms, w1 and h1, to one decimal place such that a condition can be satisfied.

", "templateType": "anything", "group": "Parallelogram", "definition": "precround(w1*h1, 1)"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Apply formulas to calculate the areas of various shapes.

"}, "duration": 0, "showstudentname": true, "name": "Jane's copy of Area of geometric shapes", "showQuestionGroupNames": false, "timing": {"allowPause": true, "timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}}, "navigation": {"reverse": true, "onleave": {"action": "none", "message": ""}, "showresultspage": "oncompletion", "browse": true, "showfrontpage": true, "allowregen": true, "preventleave": true}, "feedback": {"showtotalmark": true, "intro": "", "showactualmark": true, "showanswerstate": true, "allowrevealanswer": true, "feedbackmessages": [], "advicethreshold": 0}, "type": "exam", "contributors": [{"name": "Jane Courtney", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2154/"}], "extensions": ["geogebra", "random_person"], "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"], ["question-resources/trapezium.svg", "/srv/numbas/media/question-resources/trapezium.svg"], ["question-resources/trangle.svg", "/srv/numbas/media/question-resources/trangle.svg"], ["question-resources/parallelogram.svg", "/srv/numbas/media/question-resources/parallelogram.svg"], ["question-resources/Parallelogram_area_animated.gif", "/srv/numbas/media/question-resources/Parallelogram_area_animated.gif"], ["question-resources/rectangle_zISmvoz.svg", "/srv/numbas/media/question-resources/rectangle_zISmvoz.svg"], ["question-resources/hardertrapezium_8GqMwOo.svg", "/srv/numbas/media/question-resources/hardertrapezium_8GqMwOo.svg"], ["question-resources/Trap_advice.svg", "/srv/numbas/media/question-resources/Trap_advice.svg"], ["question-resources/Triangle_advice_lD6eKvD.svg", "/srv/numbas/media/question-resources/Triangle_advice_lD6eKvD.svg"]]}