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
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
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.