// Numbas version: exam_results_page_options {"navigation": {"browse": true, "showfrontpage": true, "showresultspage": "oncompletion", "reverse": true, "preventleave": true, "allowregen": true, "onleave": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showanswerstate": true, "showtotalmark": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "feedbackmessages": []}, "type": "exam", "name": "Volume", "timing": {"timedwarning": {"action": "none", "message": ""}, "allowPause": true, "timeout": {"action": "none", "message": ""}}, "showQuestionGroupNames": false, "showstudentname": true, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Questions involving the calculation of the volumes of shapes.

"}, "percentPass": 0, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "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, "typeendtoleave": false}, "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/"}], "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"], "metadata": {"description": "

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.

a)

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,

\\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}

b)

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}

c)

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}

d)

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}

e)

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}

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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

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Rounded theta value.

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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

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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

\\[\\mathrm{Area} = \\pi r^2.\\]

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

Calculate the area of the triangle given that the area of any triangle can be calculated using the formula

\\[\\mathrm{Area} = \\frac{1}{2}ab\\sin{C}.\\]

{geogebra_applet('https://www.geogebra.org/m/jcUJu6F4',defs)}

All lengths are in centimetres.

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

Calculate the volume of a cone given the formula for the volume of a cone is

\\[\\mathrm{Volume} = \\frac{h}{3} \\pi r^2.\\]

$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.

{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.

\\[\\mathrm{Volume}= \\frac{4}{3} \\pi r^3.\\]

$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.

Find the area of the trapezium given the formula for the area of a trapezium is

\\[\\mathrm{Area} = \\frac{1}{2}(a+b) h .\\]

{geogebra_applet('https://www.geogebra.org/m/Gtjzajb6',trap_defs)}

All lengths are given in metres.

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

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{person['name']} applies to find out how much the insurance for the car would cost, but is required to state the engine size in litres.

What is the engine size in litres?

[[0]] litres.

"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "scripts": {}, "showFeedbackIcon": true, "precisionType": "dp", "minValue": "metres_cubed", "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": false, "strictPrecision": false, "maxValue": "metres_cubed", "precision": "4", "marks": "2", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision."}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "

The specification of a second car gives the engine size in m$^3$. In order for {person['name']} to make a comparison {person['pronouns']['they']} convert{s} the engine size of the first car to cubic metres.

What is the engine size of the first car in units of m$^3$?

[[0]]m$^3$ Give your answer to 4 decimal places

"}], "advice": "

a)

The advertised engine size is $\\var{cc}$ cubic centimetres. To convert cubic centimetres to litres, we divide by $1000$.

\\[\\var{cc}\\div 1000= \\var{litres}\\text{ litres.}\\]

b)

In order to convert to cubic metres, we first note that

\\[ 1 \\text{cm} = 0.01 \\text{m.} \\]

An example of a volume of $1\\text{cm}^3$ is a cube with $1$cm sides. Converting each side into metres,

\\begin{align}
1\\text{cm}^3 &= 1\\text{cm}\\times1\\text{cm}\\times1\\text{cm} \\\\
&= 0.01\\text{m}\\times0.01\\text{m}\\times0.01\\text{m} \\\\
&= 0.000001\\text{m}^3 \\text{.}
\\end{align}

Therefore $\\var{cc}\\text{cm}^3$ is

\\[ \\var{cc} \\times 0.000001 = \\var{metres_cubed}\\text{m}^3\\text{.} \\]

", "tags": ["taxonomy"], "preamble": {"css": "", "js": ""}, "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "ungrouped_variables": ["person", "litres", "cc", "s", "metres_cubed"], "statement": "

{person['name']} is looking to buy a car. {capitalise(person['pronouns']['they'])} find{s} one advertised with an engine size of $\\var{cc}$cc.

{person['name']} recognises that 'cc' stands for units of cubic centimetres (cm$^3$) and knows the following conversions:

\n\n\n\n\n\n\n\n\n\n\n\n

$1$ m	$100$ cm
$1$ litre	$1000\\text{cm}^3$

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Convert figures for car engine sizes between cc (cm^3), litres, and m^3.

"}}, {"name": "Calculate density given mass and volume", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "advice": "

a)

We are told that the ball has a volume of $\\var{volume}\\text{cm}^3$ and a mass of $\\var{mass}$g, and we are asked to calculate the density of the box in g/cm$^3$.

The formula for density is

\\[\\begin{align} \\text{Density} &= \\frac{\\text{Mass}}{\\text{Volume}} \\\\[4pt]
&= \\frac{\\var{mass}}{\\var{volume}} \\\\[4pt]
&= \\var{density} \\\\
&= \\var{precround(density,2)}\\text{g/cm}^3\\text{.} \\\\
\\end{align}\\]

b)

Since the density of the ball is {if(density>1,'greater','smaller')} than the density of water, {person['name']}'s ball will {if(density>1,'sink','float')}.

", "statement": "

A solid object placed in water will sink if its density is greater than that of water ($1\\text{g/cm}^3$).

{person['name']}'s toy ball has a volume of $\\var{volume}\\text{cm}^3$ and a mass of $\\var{mass}$g. Whilst playing, {person['pronouns']['they']} drops {person['pronouns']['their']} ball into a pond.

", "variables": {"mass": {"name": "mass", "group": "Ungrouped variables", "definition": "random(55..65)", "templateType": "anything", "description": "

mass of the box

"}, "person": {"name": "person", "group": "Ungrouped variables", "definition": "random_person()", "templateType": "anything", "description": ""}, "density": {"name": "density", "group": "Ungrouped variables", "definition": "mass/volume", "templateType": "anything", "description": ""}, "volume": {"name": "volume", "group": "Ungrouped variables", "definition": "random(40..70 except mass)", "templateType": "anything", "description": "

Volume of box

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Calculate the density of an object given its mass and volume.

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The relationship between density, mass and volume is

\\[\\text{Density} = \\frac{\\text{Mass}}{\\text{Volume}}.\\]

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Round your answer to $3$ significant figures.

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What is the density of the ball?

[[0]] g/cm$^3$. Round your answer to $2$ decimal places.

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It floats

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It sinks

"], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "shuffleChoices": false, "prompt": "

Does {person['name']}'s ball float or sink?

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