Video: Using the Midpoint Method to Estimate Distance Travelled from a Table of Time and Speed Values

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.

02:16

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.

We’re 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 𝑖 equals one to 𝑛 of 𝑓 of π‘₯𝑖 times Ξ”π‘₯. Now here, Ξ”π‘₯ is the width of each subinterval. And that’s given by 𝑏 minus π‘Ž over 𝑛, where 𝑛 is the number of subintervals. So here, 𝑛 will be equal to four, and π‘Ž and 𝑏 are the lower and upper limits of the entire interval. π‘₯𝑖 are the midpoints of each subinterval.

Now, in this case, 𝑏 is equal to 16 and π‘Ž is equal to zero. And we know that 𝑛 is equal to four. So Ξ”π‘₯, the width of each subinterval, is 16 minus zero divided by four, which is equal to four. Now, going back to the rectangle idea, 𝑓 of π‘₯𝑖 is essentially the height of each rectangle at the midpoint of each interval. Now, since our lower limit is zero and Ξ”π‘₯ 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 it’s 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. It’s 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.

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