Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks

Question

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff's data follow. Time since start (min) 0 1530 45 60 7590 12 1199870 Speed (mph) (a) Assuming that Roger's speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour. lower estimate upper estimate miles miles (b) Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half lower estimate upper estimate miles tile (c) How often would Jeff have needed to measure Roger's speed in order to find lower and upper estimates within 0.1 mile of the actual distance he ran? Jeff would need to measure every minutes

Explanation / Answer

Solution :- Since Roger is decelerating, his velocity is decreasing, so a left-hand sum will give us an overestimate (and a right-hand one, an underestimate). To make the units correct, we convert the time intervals from 15 minutes to 1 4 of an hour when we compute the sum. For the first half-hour, we use only two intervals.

Therefore L = 1/4 [ 12+11+9] = 5.75 miles

and R = 1/4[11+9] =5 miles

Therefore, Roger traveled at least 5 miles and at most 5.75 miles

B) We take left- and right-hand sums for the entire 90-minute interval

L= 1/4 [12+11+9+9+8+7] = 14 and R = 1/4 [ 11+9+9+8+7+0] =44/4 = 11 miles

Therefore, Roger ran somewhere between 11 and 14 miles.

C) error = delta x [ fmax -fmin]

0.1 = delta x (12-0) so delta x= 1/120 = (1hour /120 minutes) (60 min. /1 hr )(1 min. / 2) =1time after .5 min = 30 seconds

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