Video: Pack 2 • Paper 2 • Question 13

Pack 2 • Paper 2 • Question 13

05:09

Video Transcript

Here is a speed-time graph for a bike. Part a) Work out an estimate for the distance the bike travelled in the first eight seconds. Use four strips of equal width. Part b) Is your answer to a) an underestimate or overestimate of the actual distance? Give a reason for your answer.

So we’re actually gonna start with part a. And what I’ve done first is I’ve actually looked at the time that we’re talking about in the question which is zero to eight seconds. And then, what I’ve done is I’ve actually split it up into four strips of equal width, each of them being two seconds wide. So what you can see is I’ve actually split it into these four parts and now what I’ve done is I’ve actually labelled them a, b, c, and d.

And we can see that a can be modelled with a triangle. And b, c, and d can each be modelled by different trapezium. So we’ve got three trapeziums and one triangle. Now, if we actually work out the area of each one of these shapes, what we’re actually gonna be doing is working out the distance. And the reason is cause if we think about speed being equal to distance over time and we can see that because actually our speed is metres per second. So metres is distance, second is time. So speed equals distance over time. So therefore, our distance can be equal to speed multiplied by time.

So therefore, if we’re gonna multiply speed by time, that’s gonna be the area beneath our graph. So we’re gonna start off with part a, so our section a which is the first strip. And the distance, therefore, in the first part, so a, is gonna be equal to a half of two multiplied by 0.5 and that’s because a triangle is half the base times the height. And we got two because it goes along two seconds on the 𝑥-axis and we’ve got 0.5 because it actually goes 0.5 metres per second up on the 𝑦-axis. So therefore, we can say that the area of part a is equal to 0.5. So therefore, the distance travelled is 0.5 metres.

Okay, great, we can move on to part b. Now, to calculate the area of b and the distance in section b, what we’re gonna need to do is work out the area of the trapezium. And we’ve actually got a formula for that and that says that the area of a trapezium is equal to a half 𝑎 plus 𝑏 multiplied by ℎ, where 𝑎 and 𝑏 are our parallel sides and ℎ is the distance between them, also the height.

One of the common mistakes here is students see that a trapezium is actually the way it is in this question and think that our height must be one of the sides that are parallel because height must be a length upwards. But no, it’s not the case. In these, the height is actually the distance between the two parallel sides.

So therefore, the area of b, so the distance travelled in section b, is gonna be equal to a half multiplied by 0.5 because 0.5 is actually the shortest parallel side plus 1.2 because that’s actually our other parallel side and I’ve actually marked that here on our graph then multiplied by two because that’s the distance between them of two seconds. So therefore, we get a distance of 1.7 metres. And this is the distance travelled in part b.

Okay, great, let’s move on to section c, our third strip. So now for our third strip, which is section c, we’re gonna have 𝑑 is equal to, so our distance is equal to a half multiplied by 1.2 plus 2.2 multiplied by two which is equal to 3.4 metres. Okay, great, so we’re gonna use the same method once more. And we’re actually gonna try and find the area of section d, so the distance travelled in section d. So therefore, for section d, so our fourth strip, 𝑑 is gonna be equal to a half multiplied by 2.2 plus 3.3 then all multiplied by two which gives us a distance of 5.5 metres.

Okay, excellent, so what we’ve done is we’ve actually found the distance travelled in each of our four parts. So then, we can add these together to actually find out the total travelled in the first four strips which gives us 11.1 metres. So therefore, the estimate for the distance travelled in the first eight seconds is 11.1 metres. Okay, now, let’s move on to part b.

So what part b is actually asking is “Is the answer to our a part an underestimate or an overestimate?” Well, why would it be either? And let’s show you why. So I’m just gonna demonstrate why you get an underestimate or an overestimate. If we look at the left, we have a curve — that’s a convex curve.

So we can see a convex curve above where we’ve actually made our trapezium. And actually, as you can see, this will be an underestimate because a trapezium is actually slightly less than actually the distance between the two points on the curve. However, on the right-hand sketch, what I’ve sketched is a concave curve. And here, you can actually see that wherever you actually picked the two points, our trapezium is actually slightly bigger than the area below the curve. So that would be an overestimate.

So therefore, using these definitions of overestimate and underestimate, what we can say is there’s a very slight overestimate because the trapeziums used are slightly larger. And this is most notable between zero and four seconds, where the trapeziums are slightly larger than the curve.

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