// Numbas version: exam_results_page_options {"type": "exam", "timing": {"timeout": {"message": "", "action": "none"}, "timedwarning": {"message": "", "action": "none"}, "allowPause": true}, "percentPass": 0, "navigation": {"showfrontpage": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "reverse": true, "allowregen": true, "onleave": {"message": "", "action": "none"}}, "duration": 0, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 1, "name": "Group", "questions": [{"name": " Using a speed graph to find distance", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "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": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "metadata": {"description": "
Use a piecewise linear graph of speed against time to find the distance travelled by a car.
\nFinally, use the total distance travelled to find the average speed.
", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": ["d2farea", "area"], "advice": "We can use a speed graph to calculate the distance travelled in a given time interval by finding the area under the line between the start and end times.
\nThe shape made by the speed curve, the line $x=0$, and the lines $t=4$ and $t=6$ seconds is a rectangle, so we can work out the area of this section by multiplying the width by the height.
\nThe rectangle is $2$ seconds wide, and $\\var{c1}$ ms-1 high.
\n\\begin{align}
\\text{Area} &= \\text{width} \\times \\text{height}\\\\
&= 2 \\times\\var{c1}\\\\
&=\\simplify{2{c1}}\\text{.}
\\end{align}
So the distance covered in this two second interval is $\\simplify{2{c1}}$ m.
\nThe shape made by the line and $x=0$ between $0$ and $2$ seconds forms a right-angled triangle with width $2$ and height $\\var{b1}$.
\n\\begin{align}
\\text{Area}&= \\frac{1}{2}\\times \\text{width} \\times \\text{height}\\\\
&= \\frac{1}{2} \\times 2 \\times \\var{b1}\\\\
&=\\var{b1} \\text{.}
\\end{align}
So therefore,the distance covered in this two second interval, and our answer, is $\\simplify{{b1}}$ meters.
\nThe shape made by the speed curve and $x=0$ between $2$ and $4$ seconds forms a trapezium. This can be broken down in to a right angle triangle (let's call this $A$) and a rectangle (we'll call this $B$).
\nTriangle $A$ has width $2$ m and height $\\var{c1}-\\var{b1}$ ms-1.
\n\\begin{align}
A &= \\frac{1}{2}\\times \\text{width} \\times \\text{height}\\\\
&= \\frac{1}{2}\\times2 \\times\\ (\\var{c1}-\\var{b1})\\\\
&= \\var{c1}-\\var{b1}\\\\
&=\\simplify{{c1}-{b1}}\\text{.}
\\end{align}
We can work out the area of the rectangle $B$ by multiplying its width, $2$ seconds, by its height, $\\var{b1}$ ms-1:
\n\\begin{align}
B &= \\text{width} \\times \\text{height}\\\\
&= 2 \\times(\\var{c1}-\\var{b1})\\\\
&=2 \\times \\simplify{{c1}-{b1}}\\\\
&=\\simplify{2{c1-b1}}\\text{.}
\\end{align}
We can now work out the whole area under the line by adding these two areas together:
\n\\begin{align}
\\text{Area} &= A + B \\\\
&=\\simplify{{c1}-{b1}} + \\simplify{2{c1-b1}} \\\\
&=\\simplify{2{c1-b1}+{c1}-{b1}} \\text{.}
\\end{align}
The distance covered in this interval is $\\var{2(c1-b1)+c1-b1}$ m.
\nSpeed is the distance travelled per unit of time.
\n\\begin{align}
\\text{speed} &= \\frac{\\text{distance}}{\\text{time}} \\\\[0.5em]
&= \\frac{\\var{area}}{10} \\\\[0.5em]
&=\\simplify[!fractionNumbers]{{area/10}} \\text{ ms}^{-1}\\text{.}
\\end{align}
You are part of an elite team analysing a high speed car race. You are given the following graph mapping the speed of one particular car as it drives around a section of the race course. The horizontal axis plots time in seconds whilst the vertical axis maps speed in metres per second ($ms^{-1}$).
\n{geogebra_applet('cecdYjwp',[[\"a1\",a1],[\"b1\",b1],[\"c1\",c1],[\"d1\",d1],[\"z1\",z1],[\"f1\",f1]])}
", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "Use the graph to calculate the distance the car travels between $4$ and $6$ seconds.
\nDistance travelled $=$ [[0]]metres
", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "c1*2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "c1*2", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "Use the graph to calculate the distance the car travels between $0$ and $2$ seconds.
\nDistance travelled $=$ [[0]]metres
", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "b1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "b1", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "Use the graph to calculate the distance the car travels between $2$ and $4$ seconds.
\nDistance travelled $=$ [[0]]metres
", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "b1+c1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "b1+c1", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "The car travelled $\\var{area}$ metres over the $10$ second period. Calculate the average speed of the the car over the $10$ seconds in metres per second. Give your answer as a whole number or a decimal to $1$ decimal place.
\nAverage speed $=$ [[0]] ms-1
", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "variableReplacements": [], "strictPrecision": true, "showCorrectAnswer": true, "mustBeReducedPC": 0, "precision": "1", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "{area/10}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{area/10}", "mustBeReduced": false, "allowFractions": false, "marks": 1, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "showPrecisionHint": true, "precisionMessage": "You have not given your answer to the correct precision."}]}], "tags": ["average speed", "calculating distance", "constant accelerations", "distance", "Distance", "graphs", "speed graph", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"d2farea": {"description": "", "group": "Ungrouped variables", "definition": "2*z1+d1+f1", "name": "d2farea", "templateType": "anything"}, "c1": {"description": "", "group": "speeds", "definition": "random(8,10,12,14)+b1", "name": "c1", "templateType": "anything"}, "a1": {"description": "", "group": "speeds", "definition": "0\n", "name": "a1", "templateType": "anything"}, "d1": {"description": "", "group": "speeds", "definition": "c1", "name": "d1", "templateType": "anything"}, "b1": {"description": "", "group": "speeds", "definition": "random(6,8,10)+a1\n", "name": "b1", "templateType": "anything"}, "area": {"description": "", "group": "Ungrouped variables", "definition": "d2farea+c1*2+b1+c1+b1", "name": "area", "templateType": "anything"}, "z1": {"description": "", "group": "speeds", "definition": "c1-random(8,10,12)", "name": "z1", "templateType": "anything"}, "mbc": {"description": "", "group": "acceleration", "definition": "(c1-b1)/2", "name": "mbc", "templateType": "anything"}, "mde": {"description": "", "group": "acceleration", "definition": "(z1-d1)/2", "name": "mde", "templateType": "anything"}, "mab": {"description": "", "group": "acceleration", "definition": "(b1-a1)/2", "name": "mab", "templateType": "anything"}, "mef": {"description": "", "group": "acceleration", "definition": "(f1-z1)/2", "name": "mef", "templateType": "anything"}, "f1": {"description": "", "group": "speeds", "definition": "2", "name": "f1", "templateType": "anything"}, "mcd": {"description": "", "group": "acceleration", "definition": "(d1-c1)/2", "name": "mcd", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Converting units of length (km/m/miles)", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}], "variable_groups": [], "variables": {"miles": {"templateType": "anything", "definition": "km*km_to_miles", "description": "", "name": "miles", "group": "Ungrouped variables"}, "location": {"templateType": "anything", "definition": "random('Boston', 'Edinburgh', 'Great Welsh', 'London')", "description": "", "name": "location", "group": "Ungrouped variables"}, "km_to_miles": {"templateType": "anything", "definition": "0.62", "description": "conversion rate km to miles
", "name": "km_to_miles", "group": "Ungrouped variables"}, "person": {"templateType": "anything", "definition": "random_person()", "description": "", "name": "person", "group": "Ungrouped variables"}, "km": {"templateType": "anything", "definition": "random(24..28#2)", "description": "", "name": "km", "group": "Ungrouped variables"}}, "type": "question", "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "km*1000", "maxValue": "km*1000", "marks": 1, "variableReplacements": []}], "prompt": "The week before the event, {person['name']} goes on a final training run. An app on {person['pronouns']['their']} phone tells {person['pronouns']['them']} that {person['pronouns']['they']} ran $\\var{km}$ km.
\nWhat was the length of {person['pronouns']['their']} training run in metres?
\n[[0]] metres.
\n\n"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "precisionType": "dp", "type": "numberentry", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "strictPrecision": false, "maxValue": "miles+0.5", "precision": 0, "marks": 1, "mustBeReduced": false, "variableReplacements": [], "minValue": "miles-0.5", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision."}], "prompt": "
Use the approximate conversion, $1$ kilometre = $0.62$ miles, to find the length of {person['pronouns']['their']} training run in miles.
\n[[0]] miles. Round your answer to the nearest mile.
"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "42-0.5", "maxValue": "42+0.5", "marks": 1, "variableReplacements": []}], "prompt": "{person['name']} knows that the {location} Marathon is 26 miles. Use the conversion rate in part b) to find the approximate length in km.
\n[[0]] km round your answer to the nearest km
"}], "advice": "{person['name']}'s training run is $\\var{km}$ km long.
\nTo convert $\\var{km}$ km into metres, we multiply $\\var{km}$ by $1000$.
\n\\[\\var{km}\\times1000= \\var{km*1000}\\text{ metres.}\\]
\n\nTo convert $\\var{km}$ km into miles, we multiply $\\var{km}$ by the conversion rate given: $0.62$.
\n\\[\\begin{align}
\\var{km}\\times\\frac{5}{8}&= \\var{miles}\\\\
&=\\var{dpformat(miles,0)}\\text{ miles, rounded to the nearest integer.}
\\end{align}\\]
The {location} Marathon is $26$ miles long. To convert to km we multiply by the inverse of the conversion rate given in part b):
\n\\[ 26 \\times \\frac{1}{0.62} = 42\\text{ miles, rounded to the nearest integer.} \\]
\n", "tags": ["taxonomy"], "preamble": {"js": "", "css": ""}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["km_to_miles", "km", "miles", "person", "location"], "statement": "
{person['name']} is training for the {location} Marathon.
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Convert from km to metres and miles, and miles to km.
"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Converting units of height (feet/inches/cm)", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"stepsPenalty": 0, "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "cm", "maxValue": "cm", "marks": "2", "variableReplacements": []}], "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "What is {person['name']}'s height in centimetres?
\n[[0]]cm round your answer to the nearest cm
", "steps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "60", "maxValue": "60", "marks": "0.5", "variableReplacements": [], "prompt": "We have information on how to convert feet to inches and inches to cm, but not feet to cm. We will therefore first convert the height into inches only.
\nWhat is $5$ft in inches?
"}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "60+inches", "maxValue": "60+inches", "marks": "0.5", "variableReplacements": [], "prompt": "What is {person['name']}'s height in inches?
"}, {"variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "We can now use the conversion rate for inches to cm to find {person['name']}'s height in cm.
"}]}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "mustBeReducedPC": 0, "showFeedbackIcon": true, "precisionType": "dp", "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "strictPrecision": false, "minValue": "cm/100", "maxValue": "cm/100", "marks": 1, "variableReplacements": [], "showPrecisionHint": false, "precision": "2", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"]}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "What is {person['name']}'s height in metres?
\n[[0]]m round your answer to 2 decimal places
\n"}], "advice": "\n\n{person['name']} is $5$ft $\\var{inches}$ inches tall.
\nTo find {person['name']}'s height into cm, we first convert it into inches.
\nWe can convert feet into inches by multiplying by $12$. So $5$ feet is
\n\\[ 5\\times12=60 \\text{ inches.}\\]
\nTherefore $5$ft $\\var{inches}$ inches is
\n\\[
60+\\var{inches} = \\var{60+inches}\\text{ inches.}
\\]
We can now convert this to cm by multiplying by $2.54$.
\n\\[\\var{60+inches}\\times2.54=\\var{cm}\\text{ cm, rounded to the nearest integer.}\\]
\nTo convert centimetres into metres, we divide by $100$:
\n\\[\\var{cm}\\div100=\\var{dpformat(cm/100,2)} \\text{ metres.}\\]
\nTherefore, {person['name']} is $\\var{dpformat(cm/100,2)}$ metres tall.
\n\n", "tags": ["taxonomy"], "variables": {"cm": {"templateType": "anything", "description": "", "definition": "precround((inchfeet*2.54),0)", "name": "cm", "group": "Ungrouped variables"}, "inchfeet": {"templateType": "anything", "description": "", "definition": "(5*12)+inches", "name": "inchfeet", "group": "Ungrouped variables"}, "person": {"templateType": "anything", "description": "", "definition": "random_person()", "name": "person", "group": "Ungrouped variables"}, "s": {"templateType": "anything", "description": "", "definition": "if(person['gender']='neutral','','s')", "name": "s", "group": "Ungrouped variables"}, "inches": {"templateType": "anything", "description": "", "definition": "random(4..11)", "name": "inches", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["person", "inches", "inchfeet", "cm", "s"], "statement": "{person['name']} is $5$ft $\\var{inches}$ inches tall and would like to find {person['pronouns']['their']} height in cm.
\n{capitalise(person['pronouns']['they'])} find{s} the following unit conversion table:
\n$1$ foot | \n$12$ inches | \n
$1$ inch | \n$2.54$ cm | \n
$1$ metre | \n$100$ cm | \n
Convert a height given in feet and inches into cm and then metres.
"}, "variablesTest": {"condition": "", "maxRuns": "1000"}}, {"name": "Using compound units - speed", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [], "precisionMessage": "Round your answer to $2$ decimal places.
", "precisionType": "dp", "mustBeReducedPC": 0, "precisionPartialCredit": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": false, "strictPrecision": false, "maxValue": "distance/seconds", "precision": "2", "marks": 1, "scripts": {}, "minValue": "distance/seconds", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"]}], "variableReplacements": [], "showFeedbackIcon": true, "prompt": "What was the runner's average speed, in metres per second?
\n[[0]] m/s
\n"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [], "precisionMessage": "
Round your answer to $2$ decimal places.
", "precisionType": "dp", "mustBeReducedPC": 0, "precisionPartialCredit": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": false, "strictPrecision": false, "maxValue": "distance/seconds*3.6", "precision": "2", "marks": 1, "scripts": {}, "minValue": "distance/seconds*3.6", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"]}], "variableReplacements": [], "showFeedbackIcon": true, "prompt": "How fast is this in kilometres per hour?
\n[[0]] km/h
"}], "advice": "To find the average speed of the runner in meters per second (m/s), we divide the distance covered by the runner (in metres) by the time taken for the runner to run this distance (in seconds).
\n\\[
\\begin{align}
\\text{Average speed} &= \\displaystyle\\frac{\\var{distance}}{\\var{seconds}}\\\\
&= \\var{distance/seconds}\\\\
&= \\var{dpformat(distance/seconds,2)}\\; \\text{m/s} \\; (\\text{rounded to $2$ decimal places}).
\\end{align}
\\]
We can convert the average speed of the runner that we calculated in a) in metres per second to kilometres per hour using the following two equivalences:
\n\\[1\\text{m} = \\displaystyle\\frac{1}{1000}\\text{km},\\]
\n\\[
1 \\; \\text{second} = \\displaystyle\\frac{1}{60} \\; \\text{minutes} = \\displaystyle\\frac{1}{3600} \\; \\text{hours}.
\\]
We know from a) that the average speed of the runner in m/s was $\\var{dpformat(distance/seconds,5)}$ m/s ($5$ d.p), so to convert this speed to km/h we first need to convert metres to kilometres,
\n\\[\\var{dpformat(distance/seconds,5)} \\; \\text{m/s} = \\var{dpformat(distance/seconds/1000,5)} \\text{km/s} \\; (5 \\; \\text{d.p})\\]
\nThen we convert seconds to hours,
\n\\[1 \\; \\text{second} = \\displaystyle\\frac{1}{3600} \\; \\text{hours}.\\]
\nNow we have
\n\\[\\var{dpformat(distance/seconds,5)} \\; \\text{m/s} = \\var{sigformat(distance/seconds/1000,5)} \\; \\text{kilometres per $\\displaystyle\\frac{1}{3600}$ hours}.\\]
\nWe want a rate per one hour, so we multiply by $3600$ to obtain a measurement in km/h:
\n\\[\\begin{align}\\var{sigformat(distance/seconds/1000,5)} \\; \\text{kilometres per $\\displaystyle\\frac{1}{3600}$ hours} &= \\var{siground(distance/seconds/1000,5)*3600} \\; \\text{km}/\\text{h}\\\\&=\\var{dpformat(distance/seconds*3.6, 2)} \\; \\text{km}/\\text{h} \\; (\\text{rounded to $2$ decimal places}).\\end{align}\\]
\n\\[\\var{sigformat(distance/seconds/1000,5)} \\; \\text{kilometres per $\\displaystyle\\frac{1}{3600}$ hours} = \\var{distance/seconds*3.6} \\; \\text{km}/\\text{h}.\\]
\nNote that throughout this calculation we have rounded all figures to $5$ decimal places for convenience; when doing calculations which involve long decimals, you should always input the full figure into your calculator to avoid getting an incorrect answer due to rounding.
", "tags": ["taxonomy"], "variables": {"speed": {"templateType": "anything", "description": "Athlete's speed, in m/s.
\n4m/s is about 9 mph, a bit faster than a jog. The current world record is 12m/s.
", "definition": "random(4..8#0)", "name": "speed", "group": "Ungrouped variables"}, "distance": {"templateType": "anything", "description": "Distance that the runner ran.
", "definition": "random(80,100,150,200)", "name": "distance", "group": "Ungrouped variables"}, "seconds": {"templateType": "anything", "description": "Time taken to cover the distance, in seconds.
", "definition": "floor(distance/speed)", "name": "seconds", "group": "Ungrouped variables"}}, "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": "100"}, "functions": {}, "ungrouped_variables": ["distance", "seconds", "speed"], "statement": "An athlete runs $\\var{distance}$ m in $\\var{seconds}$ seconds.
\nRound each of your answers to two decimal places.
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Calculate a speed in m/s given distance and time taken, then convert that to km/hour
"}}, {"name": "Use speed and distance to calculate time ", "extensions": [], "custom_part_types": [], "resources": [], "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": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "metadata": {"description": "Calculate the time taken for a certain distance to be travelled given the average speed and the distance travelled.
\nSmall, simple question.
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["kilometres", "speed"], "type": "question", "rulesets": {}, "variable_groups": [], "statement": "At a greyhound race, a fake rabbit moves around the inside of the track to motivate the dogs to run.
\nThe track is $\\var{kilometres}$km long and the rabbit moves at a constant speed of $\\var{speed}$m/s.
", "advice": "We are told that the track is $\\var{kilometres}$km long and that the speed of the rabbit is $\\var{speed}$m/s (metres per second).
\nMost people remember the relationship between speed, distance and time with the formula
\n\\[\\text{Average speed} = \\frac{\\text{Distance travelled}}{\\text{Total time taken}}.\\]
\nWe can rearrange the formula for average speed to give us the formula for the time taken:
\n\\[\\text{Total time taken} = \\displaystyle\\frac{\\text{Distance travelled}}{\\text{Average speed}}.\\]
\nFirstly, we must convert the units of the length of the race from kilometres to metres. We know that $1\\text{km} = 1000\\text{m}$, therefore
\n\\begin{align}
\\var{kilometres}\\text{km} &= (\\var{kilometres} \\times 1000)\\text{m}\\\\
&= \\var{{kilometres}*1000}\\text{m}.
\\end{align}
Therefore, the time taken for the rabbit to finish one lap is
\n\\[\\displaystyle\\frac{\\var{{kilometres}*1000}}{\\var{speed}} = \\var{({kilometres}*1000)/{speed}} \\; \\text{seconds}.\\]
\n\\[ \\begin{align} \\displaystyle\\frac{\\var{{kilometres}*1000}}{\\var{speed}} &= \\var{({kilometres}*1000)/{speed}} \\; \\text{seconds}\\\\ &= \\var{dpformat(({kilometres}*1000)/{speed}, 0)} \\; \\text{seconds} \\; (\\text{to the nearest second}). \\end{align}\\]
", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "stepsPenalty": 0, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "prompt": "Speed, distance and time are related by the equation
\n\\[\\text{Average speed} = \\frac{\\text{Distance travelled}}{\\text{Total time taken}}.\\]
", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0}], "prompt": "Calculate the time taken for the rabbit to complete a circuit of the track.
\n[[0]] seconds Round your answer to the nearest second.
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"}], "showFeedbackIcon": true, "marks": 0, "showCorrectAnswer": true}], "tags": ["compound measures", "Compound measures", "Compound units", "compound units", "Distance", "distance", "km", "metres per second", "speed", "taxonomy", "Time", "time"], "preamble": {"js": "", "css": ""}, "functions": {}, "variables": {"speed": {"description": "Speed of the greyhound in metres per second.
", "group": "Ungrouped variables", "definition": "random(14..18)", "name": "speed", "templateType": "anything"}, "kilometres": {"description": "Kilometres to be ran by the greyhound
", "group": "Ungrouped variables", "definition": "random(0.35..0.49 #0.01)", "name": "kilometres", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Straight line equation application: measuring sunflower height", "extensions": ["jsxgraph", "random_person"], "custom_part_types": [], "resources": [], "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": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "metadata": {"description": "An applied example of the use of two points on a graph to develop a straight line function, then use the t estimate and predict. MCQ's are also used to develop student understanding of the uses of gradient and intercepts as well as the limitations of prediction.
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\nObservation | \nA | \nB | \n
Day | \n$\\var{xa}$ | \n$\\var{xb}$ | \n
height (cm) | \n$\\var{ya}$ | \n$\\var{yb}$ | \n
{person['name']} plots the 2 points:
\n{plotPoints()}
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", "templateType": "anything", "definition": "random(3..5)"}}, "tags": ["assessing the accuracy of a graph", "estimations", "gradient", "interpreting graphs", "interpreting the gradient", "limitations of a line equation based on data used to create the equation", "line equation", "Straight Line", "straight line", "taxonomy", "using graphs to estimate a y value", "y-intercept"], "ungrouped_variables": ["m", "c", "d"], "functions": {"advicePoints": {"language": "javascript", "type": "html", "parameters": [], "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[-1,yb+5,xb+3,-2],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n\nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-2,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: true, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\n\n\nquestion.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\nvar expr = question.parts[2].gaps[0].display.studentAnswer();\n//define ans as this \ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;"}, "plotPoints": {"language": "javascript", "type": "html", "parameters": [], "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[-1,yb+5,xb+3,-2],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n var ans;\n var nscope = new Numbas.jme.Scope([scope,{variables:{x:new Numbas.jme.types.TNum(0)}}]);\n//this is the beating heart of whatever plots the function,\n//I've changed this from being curve to functiongraph\n var line = board.create('functiongraph',[function(x){\nif(ans) {\n try {\nnscope.variables.x.value = x;\n var val = Numbas.jme.evaluate(ans,nscope).value;\n return val;\n }\n catch(e) {\nreturn 163;\n }\n}\nelse\n return 163;\n },-2,22]\n , {strokeColor:\"blue\",strokeWidth: 4} );\n \nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-2,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: false, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\nquestion.lines = {\n l:line, c:correct_line\n}\n\n question.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\nvar expr = question.parts[2].gaps[0].display.studentAnswer();\n\n//define ans as this \ntry {\n ans = Numbas.jme.compile(expr,scope);\n}\ncatch(e) {\n ans = null;\n}\nline.updateCurve();\ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "advice": "The gradient is the ratio of vertical change ($y_2-y_1$) to horizontal change ($x_2-x_1$).
\n\\[ m = \\frac{y_2-y_1}{x_2-x_1}=\\frac{\\simplify[!collectNumbers]{{yb}-{ya}}}{\\simplify[!collectNumbers]{{xb}-{xa}}}=\\frac{\\simplify{{yb}-{ya}}}{\\simplify{{xb}-{xa}}}=\\simplify[simplifyFractions,unitDenominator]{({yb-ya})/({xb-xa})}\\text{.}\\]
\nRearranging the equation $y=mx+c$ for $c$ and using point A:
\n\\[ c = y_1-mx_1 = \\var{ya}-\\var{m}\\times\\var{xa}=\\simplify{{ya-m*xa}}\\text{.}\\]
\nWe then check this against point $B$:
\n\\[ y_2 = mx_2 + c = \\simplify[fractionNumbers]{{m}{xb}+{c}}=\\simplify{{m}*{xb}+{c}}\\text{.}\\]
\nWe now substitute the values for $m$ and $c$ into the equation of a straight line, $y=mx+c$,
\n\\[y=\\simplify[!noLeadingMinus,unitFactor]{{m} x+ {c}}\\text{.}\\]
\n{advicePoints()}
\nThe gradient represents the vertical change (height in cm) per unit of the horizontal axis (days): the change in height of the sunflower per day.
\nSubstituting $x=\\var{d}$ into the straight line equation, the height $y$ after $\\var{d}$ days is
\n\\begin{align}
y&=\\simplify{{m}}x+\\var{c}\\\\
&=\\simplify[]{{m}{d}}+\\var{c}\\\\
&=\\var{m*d+c}\\text{cm.}
\\end{align}
Substituting $x=\\var{1826}$ into the straight line equation, the height $y$ after $1826$ days is
\n\\begin{align}
y&=\\simplify{{m}}x+\\var{c}\\\\
&=\\simplify[]{{m}1826} + \\var{c}\\\\
&=\\var{m*1826+c}\\text{cm.}
\\end{align}
Note that this is $\\var{(m*1826+c)/100}$ metres. In 2014, a sunflower of $9.17$ metres was entered into the Guinness World Records as tallest sunflower.
\nPossible reasons that the prediction will not be accurate are:
\nInvalid reasons that the prediction will not be accurate are:
\nWhat is the gradient, $m$, of the straight line between the two points?
\n$m =$ [[0]]
", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "variableReplacements": [], "unitTests": [], "useCustomName": false, "customName": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "unitTests": [], "useCustomName": false, "minValue": "c", "correctAnswerFraction": false, "customName": "", "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "c", "showCorrectAnswer": true, "type": "numberentry", "showFractionHint": true, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "marks": 1, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}], "type": "gapfill", "prompt": "Use the gradient and the coordinates of the two points to find the height of the sunflower when {person['name']} received it.
\n[[0]] cm.
", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"scripts": {"mark": {"order": "after", "script": "console.log(this.question.lines.c)\nthis.question.lines.l.setAttribute({strokeColor: this.credit==1 ? 'green' : 'red'});\nthis.question.lines.c.setAttribute({visible: this.credit==1 ? false : true});\n"}}, "variableReplacements": [], "unitTests": [], "useCustomName": false, "customName": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"valuegenerators": [{"name": "x", "value": ""}], "variableReplacements": [], "unitTests": [], "useCustomName": false, "showPreview": true, "variableReplacementStrategy": "originalfirst", "customName": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "marks": 1, "customMarkingAlgorithm": "", "answer": "{m}*x+{c}", "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5}], "type": "gapfill", "prompt": "Let $y$ be the sunflower height and $x$ the time, in days, since {person['name']} received the sunflower. What is the equation of the straight line between the points?
\n$y(x) = $ [[0]]
\nUse the preview on {person['name']}'s plot to check your answer.
", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"marks": 0, "displayType": "radiogroup", "unitTests": [], "choices": ["The length of time taken in days for the sunflower to grow $1$ cm
", "The change in height (in cm) of the sunflower over $1$ day
", "The width of the ruler used to measure the sunflower
", "All of the above
"], "scripts": {}, "distractors": ["", "", "", ""], "shuffleChoices": false, "prompt": "What does the gradient represent?
", "showCellAnswerState": true, "useCustomName": false, "minMarks": 0, "variableReplacements": [], "maxMarks": 0, "customName": "", "extendBaseMarkingAlgorithm": true, "matrix": [0, "1", 0, 0], "displayColumns": "1", "showCorrectAnswer": true, "type": "1_n_2", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "variableReplacements": [], "unitTests": [], "useCustomName": false, "customName": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "unitTests": [], "useCustomName": false, "minValue": "d*m+c", "correctAnswerFraction": false, "customName": "", "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "d*m+c", "showCorrectAnswer": true, "type": "numberentry", "showFractionHint": true, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "marks": 1, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}], "type": "gapfill", "prompt": "{person['name']} uses the straight line equation to predict the future height of the sunflower. What will the height be on day $\\var{d}$?
\n[[0]] cm
", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "variableReplacements": [], "unitTests": [], "useCustomName": false, "customName": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "unitTests": [], "useCustomName": false, "minValue": "1826*m+c", "correctAnswerFraction": false, "customName": "", "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "1826*m+c", "showCorrectAnswer": true, "type": "numberentry", "showFractionHint": true, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "marks": 1, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true}], "type": "gapfill", "prompt": "{person['name']} wonders if {person['pronouns']['they']} can guess what the height of the sunflower will be on {person['pronouns']['their']} 35th birthday. {capitalise(person['pronouns']['they'])} work{s} out that this is day 1826. Using the straight line equation, what would the height be on day 1826?
\n[[0]] cm
", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"variableReplacementStrategy": "originalfirst", "displayType": "checkbox", "unitTests": [], "choices": ["Sunflower height as a function of time may not have a straight linear relationship.
", "The observations only span a very limited time range.
", "There are multiple straight linear relationships that could be obtained using the same $2$ data points.
", "Sunflower height never actually increases over time.
"], "scripts": {}, "distractors": ["", "", "", ""], "shuffleChoices": false, "marks": 0, "showCellAnswerState": true, "useCustomName": false, "minMarks": 0, "variableReplacements": [], "maxMarks": "1", "prompt": "{person['name']} doubts {person['pronouns']['their']} result. Which of the following reason(s) may mean that the height on {person['pronouns']['their']} 35th birthday is not accurate?
", "customName": "", "warningType": "none", "extendBaseMarkingAlgorithm": true, "matrix": ["0.5", "0.5", "-0.5", "-0.5"], "displayColumns": "1", "showCorrectAnswer": true, "type": "m_n_2", "maxAnswers": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "minAnswers": 0}]}]}], "feedback": {"showactualmark": true, "intro": "", "allowrevealanswer": true, "advicethreshold": 0, "showtotalmark": true, "showanswerstate": true, "feedbackmessages": []}, "name": "Linear measure - distance and length", "metadata": {"description": "Some questions to do with measures of distance and length.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "showstudentname": true, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": ["geogebra", "jsxgraph", "random_person"], "custom_part_types": [], "resources": []}