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

Question

2.

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 follows Time since start (min) 15 30 45 60 75 90 Speed (mph) 11 10 8 86 5 0 (a) Assuming that Roger's speed is never increasing, give upper and lower estimates for the distance Roger ran during the first halfhour. 3 X miles (lower estimate) miles (upper estimate) (b) 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

Explanation / Answer

1a) Max speed in first half hour was 11mph so in 0.5 hr , he can at max cover , 11*0.5 = 5.5 miles (maximum)

similarly , minimum speed in first half hour is 8 mph , so minimum distance = 8*0.5 = 4 miles (minimum)

b)following similar logic ,

max = 11*1.5 = 16.5 miles (max )

min = 5 *1.5 = 7.5 miles (min)

c)total miles covered in actual = 11*0.25 + 10*0.25 +8*0.25 +8*0.25 +6*0.25 +5*0.25 =12 miles

total time taken = 90 minutes

12 miles takes 90 min so how many minutes does 0.1 mile take ?

ans = 90*0.1/12 = 3/4 so every 3/4th minute =0.75 minute -------answer

2) (t,v) coordinate system

first point = (0,5) , second point (2,37) third point = (4,77) , fourth point = (6,133)

distance is the area under these lines , they form triangles so , calculate 0.5 *base *height

dist = 0.5*2 *32 +0.5*2*40+0.5*2*56 = 128 -------ANSWER

b ) 4th option , it is an underestimate because the velocity function is concave up

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