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) 0 15 30 45 60 75 90 Speed (mph) 13 12 10 10 8 7 0 (a) Assuming that Roger's speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour. miles (lower estimate) miles (upper estimate)

Answer:Lower estimate 5,5 miles Upper estimate 6,25

Explanation:

First let's consider that 15 minutes is equal to 15/60 hours, that is 1/4=0,25 hours, we need this since the speed is given in miles per hour, then the time should be in hours too.

For the upper estimate, we assume the speed is constant on the highest possible value, that is the one at the start of the interval, for example from 0 to 15, there are two possible speeds 13 or 12 mph, for the upper estimate we consider the 13 and for the lower we consider the 12 a follows.

Upper estimate:

Speed_0-15 = 13 mph then the distance_0-15=13 mph*1/4 hours=3,25 miles

Speed_15-30=12 mph then the distance_15-30=12 mph*1/4 hours=3 miles

Then Distance_upper-estimate = 3,25 + 3 = 6,25 miles

Lower estimate:

Speed_0-15 = 12 mph then the distance_0-15=12 mph*1/4 hours=3 miles

Speed_15-30=10 mph then the distance_15-30=10 mph*1/4 hours=2,5 miles

Then Distance_upper-estimate = 3 + 2,5 = 5,5 miles

All this because we consider the distance traveled X at constant speed is given by X=v*t, v:=speed, t:=time