Suppose the odometer on our car is broken and we want to estimate the distance d

Question

Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30 second time interval. We take the speedometer readings every five seconds and record them in the following table: In order to have the time and the velocity in consistent units, let's convert the velocity readings to feet per second (1 mi/h = 5280/3600 ft/s) During the first five seconds the velocity doesn't change very much, so we can estimate the distance traveled during that time by assuming that the velocity is constant. If we take the velocity during that time interval to be the initial velocity (23 ft/s), then we obtain the approximate distance traveled during the first five seconds: 23 ft/s x 5 s = ft Similarly, during the second time interval the velocity is approximately constant and we take it to be the velocity when t = 5 s. So our estimate for the distance traveled from t = 5 s to t = 10 s is 31 ft/s x 5 s = ft If we add similar estimates for the other time intervals, we obtain an estimate for the total distance traveled: (23 times 5) + (31 times 5) + (32 times 5) + (40 times 5) + (48 times 5) + (44 times 5) = ft We could just as well have used the velocity at the end of each time period instead of the velocity at the beginning as our assumed constant velocity. Then our estimate becomes (31 times 5) + (32 times 5) + (40 times 5) + (48 times 5) + (44 times 5) + (43 times 5) = ft If we had wanted a more accurate estimate, we could have taken velocity readings every two seconds, or even every second.

Explanation / Answer

1 mile/hr = 5280/3600 or 22/15 ft/sec

So, 21 mi/h = 21*(22/15) = 31 ---> first answer

27 mi/h = 27*22/15 = 40 ---> second answer

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23 ft/s * 5 s = 115 ft ---> third answer

31 ft/s * 5 s = 155 ft ---> fourth answer

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1090 ft --> fifth answer

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1190 ft ----> sixth answer

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