The average round trip speed S in miles per hour of a vehicle traveling a distance of d miles in each direction is given by:

Question

The average round trip speed S in miles per hour of a vehicle traveling a distance of d miles in each direction is given by:

s=2d/(d/r_1 +d/r_2 )

where r1 and r2 are the rates of the speed for the initial trip and the return trip respectively. Suppose a motorist travels 200 miles to an event and averages 50 mph for the trip of the event. Determine the average speed necessary for the return trip if the motorist wants the average speed for the round trip to be at least 55 mph. Round to one decimal place. Write a sentence that explains your result.

Answer 1

Ally,

For the overall average to be 55, we need to solve the following equation:

 

55 = 400/t

 

because the average speed of a trip is just total distance / total time. So solving for t we get:

 

t = 400/55 = 7.27 hours

 

Now if the trip TO the event took t₁ = 200/50 = 4 hours, then the return trip needs to happen in:

 

7.27 - 4 = 3.27 hours

 

So the speed necessary to achieve that is

 

s = 200/3.27 = 61.11 mph  You may have expected it to be 60, because the average of 50 and 60 is 55. BUT, they spend less time at the higher speed, so it is essentially a weighted average. The greater time spent at 50 mph gives it greater weight. Less time at the higher peed means that it must be even higher to achieve the same weight.