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

Question

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and dlocks 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 follows Time since start | 0 | 15| 301 451 60 75 10 8 8 7 6 (a) Assuming that Rogers speed is never increasing, give upper and lower estimates for the distance Roger ran during the first halr hour. 9.75 12.5x miles (upper estimate) x miles (lower estimate) Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half. miles (lower estimate) miles (upper estimate) (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? every minutes

Explanation / Answer

Notice that the speeds are decreasing :

So, for lower estimate, we simply find right hand sum

Considering t = 0 , 15 and 30,
we take the values 10 and 8

10 + 8 = 18 mph

Now, delta(t) = 15 min = 1/4

18 * 1/4

4.5 miles ---> ANS

Upper estimate :
Take 11 and 10

11 + 10 ---> 21 mph

21 * 1/4

5.25 miles ---> ANS

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b)
Lower :
10 + 8 + 8 + 7 + 6 + 0
39

39*1/4

9.75 miles ---> ANS

Upper :
11 + 10 + 8 + 8 + 7 + 6
50

50 * 1/4

12.5 miles ---> ANS

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