### Video Transcript

The table shows the speed of a car,
in feet per second, at various times during a 16-second interval. Estimate the distance the car
travels using the midpoint method with four subintervals.

Weβre given a speed time graph. We can find an estimate for the
distance by looking at the area underneath the curve. The midpoint method is a way of
estimating the area by splitting the region into rectangles. The formula is the area is
approximately equal to the sum from π equals one to π of π of π₯π times Ξπ₯. Now here, Ξπ₯ is the width of each
subinterval. And thatβs given by π minus π
over π, where π is the number of subintervals. So here, π will be equal to four,
and π and π are the lower and upper limits of the entire interval. π₯π are the midpoints of each
subinterval.

Now, in this case, π is equal to
16 and π is equal to zero. And we know that π is equal to
four. So Ξπ₯, the width of each
subinterval, is 16 minus zero divided by four, which is equal to four. Now, going back to the rectangle
idea, π of π₯π is essentially the height of each rectangle at the midpoint of each
interval. Now, since our lower limit is zero
and Ξπ₯ is equal to four, our first interval is zero to four and the midpoint of
zero and four is two. The value of the function at this
point β in other words, the speed β is eight. So the first rectangle has an area
of eight times four. Our next interval, given that itβs
four units wide, is from four to eight. The midpoint of four and eight is
six. And the speed at this point is
12. So the area of our second rectangle
is 12 times four.

Our next interval is from eight,
and at this time it goes up to 12. The midpoint of eight and 12 is
10. The speed at 10 seconds is 15 or 15
feet per second. So the area of the third rectangle
is 15 times four. Our fourth rectangle goes from 12
to 16. The midpoint of 12 and 16 is
14. So the value of the function at
this point is 17. And the area of our fourth
rectangle is 17 times four. An estimate for the area is the sum
of these values. Itβs 32 plus 48 plus 60 plus 68
which is 208. Remember, we said that this area is
an estimate for the total distance travelled during the 16-second interval. Since time is in seconds and speed
is in feet per seconds, the area or the distance is in terms of feet. So the answer is 208 feet.