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Convert from km to metres, km/h to m/s, rounding to whole numbers and sig fig.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{person['name']} is training for the {location} Marathon.
", "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.}\\]
\nRounding to the nearest whole kilometer means the app would round anything $\\var{km*1000-500}\\,$m or higher up to $\\var{km}\\,$km, and anything up to $\\var{km*1000+499}\\,$m would be rounded down to ${\\var{km}\\,$km.
\n{person['name']}'s training run lasted $2\\,$hours and $\\var{mins}\\,$minutes.
\nTo convert $2\\,$hours to minutes, we multiply by $60$. To convert these minutes to seconds, we multiply by $60$ again:
\n\\begin{align}
2\\,\\text{hours}\\times60&=120\\,\\text{minutes}\\\\
120\\,\\text{minutes}\\times60&=7200\\,\\text{seconds}
\\end{align}
and to convert $\\var{mins}\\,$minutes to seconds, we multiply by $60$:
\n\\[\\var{mins}\\,\\text{minutes}\\times60=\\var{mins*60}\\,\\text{seconds}\\]
\nGiving us a total of $7200+\\var{mins*60}=\\var{7200+mins*60}\\,$seconds.
\nSince $\\var{mins}\\,$minutes was given as being correct to the nearest minute, we know that it in fact represents possible times from $\\var{mins-1}\\,$minutes and $30$ seconds to $\\var{mins}\\,$minutes and $29\\,$seconds, so in fact our total should be between $\\var{(120+{mins})*60-30}$ and $\\var{(120+{mins})*60+29}\\,$seconds.
\nTo calculate a running pace in metres per second, we simply use:
\n\\[\\text{Speed}=\\frac{\\text{Distance}}{\\text{Time}}\\]
\nOur greatest value for speed will involve the longest distance in the shortest time; and our lowest value for speed will involve the shortest distance in the longest time.
\nHighest pace:
\n\\[\\frac{\\var{({km}*1000+499)}\\,\\text{metres}}{\\var{((120+{mins})*60-30)}\\,\\text{seconds}}=\\var{bestpace}\\,\\text{m/s}\\]
\nLowest pace:
\n\\[\\frac{\\var{({km}*1000-500)}\\,\\text{metres}}{\\var{((120+{mins})*60+29)}\\,\\text{seconds}}=\\var{worstpace}\\,\\text{m/s}\\]
\nIf they can maintain a pace of $\\var{bestpace}\\,$m/s, then their race time will be:
\n\\[\\text{Time}=\\frac{\\text{Distance}}{\\text{Speed}}=\\frac{42195\\,\\text{metres}}{\\var{bestpace}\\,\\text{m/s}}=\\var{bestseconds}\\,\\text{seconds}\\]
\nIf we divide by $60$, we will get the number of minutes; dividing by $60$ again gets us the number of hours:
\n\\[\\frac{\\var{bestseconds}\\,\\text{seconds}}{60\\times60}=\\var{dechours}\\,\\text{hours}=\\var{besthours}\\,\\text{hours and }\\var{decmins}\\,\\text{mins}\\]
\n(Don't forget that $\\var{dechours-besthours}$ hours means $\\var{dechours-besthours}\\times60\\,$minutes, not $\\var{(dechours-besthours)*100}\\,$minutes.)
\nFirst convert to minutes and seconds by multiplying by $60$:
\n\\[2\\,\\text{hours and }\\var{recordmins}\\,\\text{minutes}=\\var{120+recordmins}\\,\\text{minutes}\\times60=\\var{(120+recordmins)*60}\\,\\text{seconds}\\]
\nSo the impressive runner will cover the whole course in this length of time. Meanwhile, {person['name']}'s best pace is $\\var{bestpace}\\,$m/s. So during this time they could cover a distance of $\\var{(120+recordmins)*60}\\times\\var{bestpace}=\\var{(120+recordmins)*60*bestpace}\\,$metres.
\nThis means that {person['name']} needs a head start of $42195-\\var{(120+recordmins)*60*bestpace}=\\var{42195-(120+recordmins)*60*bestpace}\\,$metres or $\\var{headstart}\\,$km.
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\nWhat was the minimum length, in metres, of {person['pronouns']['their']} training run?
\n[[0]] metres.
\nWhat was the maximum length, in metres, of {person['pronouns']['their']} training run?
\n[[1]] metres.
\nGive your answers correct to the nearest whole number.
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\nWhat is the shortest time, in seconds, for {person['name']}'s run?
\n[[0]] seconds
\nWhat is the longest time, in seconds, for {person['name']}'s run?
\n[[1]] seconds
\nGive your answers correct to the nearest whole number.
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\nWhat is the highest possible value of their running pace?
\n[[0]] m/s
\nWhat is the lowest possible value of their running pace?
\n[[1]] m/s
\nGive your answers correct to two places of decimals.
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\nAssuming they can maintain their best pace from the training run, what is the best possible time in which they can complete the race?
\n[[0]] hours and [[1]] minutes.
\nGive your answers correct to the nearest whole numbers.
An impressive time (though not a record) in the men's marathon might be $2$ hours and $\\var{recordmins}$ minutes.
\nWhat is this pace in metres per second?
\n[[0]] m/s
\nGive your answer correct to two places of decimals.
\nWhat is the smallest distance 'head start' that {person['name']} would require in order not to be beaten by such an impressive runner?
\n[[1]] km
\nGive your answer correct to one place of decimals.
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