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Video: Writing and Evaluating a Linear Function in a Real-World Context

Tim Burnham

A cyclist leaves a town and travels at a constant speed. The table below shows the distance travelled against time. Use it to find the cyclist’s speed and the distance he travelled in 630 minutes.

04:16

Video Transcript

A cyclist leaves a town and travels at a constant speed. The table shows the distance they travelled against time. Use it to find the cyclist’s speed and the distance they travelled in six hundred and thirty minutes.

And we’ve got a function table that shows us that they travelled sixty kilometres in three hours, a hundred and twenty kilometres in six hours, a hundred and eighty kilometres in nine hours, and two hundred and forty kilometres in twelve hours.

So in terms of our function, the distance travelled is the input and the output is the time that it took to travel that distance. And we can see that for the numbers we’ve been given, and we’ve been given increments of sixty, so sixty, hundred and twenty, hundred and eighty, two hundred and forty. Those amounts are going up by sixty each time.

And we can also see that every time we increase the distance by sixty kilometres, the time goes up by three hours. So in every three hours, they’re travelling sixty kilometres. So let’s write that down. Sixty kilometres takes three hours. Now we’re looking to find the cyclist’s speed. And you’d normally measure that, in this case in, kilometres per hour. So we want to know how far they would travel in one hour.

Now in order to turn three hours into one hour, I need to divide by three. So we’re looking for a third the amount of time. And in a third the amount of time, travelling at a constant speed, we’re gonna go a third the distance. So I’m going to divide that distance by three.

And that tells me that I would travel twenty kilometres in one hour, so my speed is twenty kilometres per hour. Now thinking about the rule for this function table, if we take the input say sixty kilometres, how do we convert that into the output of three hours? Well we have to divide it by twenty. And likewise to turn a hundred and twenty into six, I divide by twenty. And the same for the other bits of data too.

So if we call our input 𝑥 and our output 𝑦, then our function rule is 𝑦 is equal to 𝑥 divided by twenty or 𝑥 over twenty or even one twentieth of 𝑥.

Now the second part of the question asks us how far they travel in a time of six hundred and thirty minutes. Well six hundred and thirty is six hundred minutes plus thirty minutes.

And the reason I split it up that way was because six hundred minutes is exactly ten hours; ten times sixty’s six hundred. And that leaves us with thirty minutes, nought point five hours. So we’ve got a 𝑦-value of ten point five hours, and we want to find the corresponding 𝑥-value, the distance, that they travelled in that ten and a half hours.

So we’ve put the ten point five in for the 𝑦-value, the time, and that’s equal to one twentieth of 𝑥. Now if I multiply both sides of my equation by twenty, then on the right-hand side I’ve got twenty times a twentieth, which is one. So that’s just one 𝑥 on the right-hand side. And twenty times ten point five or two times ten point five would be twenty-one, and if I multiply that by ten I’ve got two hundred and ten.

So in ten and half hours, I’ll be travelling two hundred and ten kilometres. And my answers are twenty kilometres an hour for the speed and two hundred and ten kilometres in six hundred and thirty minutes.

Now just finally before I go, I’m gonna show you another way to work out that second part of the answer. Rather than using the function rule as we did, we could’ve just used the information that we gathered earlier; if we travel twenty kilometres in one hour, then we want to know how many kilometres we travel in ten point five hours. So what do I have to do to one to turn it into ten point five? Well I have to multiply by ten point five.

So if I travel ten point five times as long at that constant speed, then I’m gonna go ten point five times as far. So I’m gonna do twenty times ten point five kilometres. And luckily, that also gives us two hundred and ten kilometres.