For speeds between 40 mph and 65 mph, a diesel truck gets 370/x miles per gallon

Question

For speeds between 40 mph and 65 mph, a diesel truck gets 370/x miles per gallon when driven at a constant speed of xx miles per hour. Diesel gasoline costs 4.5 dollars per gallon. The driver is paid 22 dollars per hour.

A) What is the total cost to have the truck driven 200 miles at a constant speed of 60 miles per hour?

B) What is the best constant speed to drive the truck to minimize the total cost to drive it 200 miles?

C) How much money is saved, per 200 miles, by driving at the the constant speed you found in problem B) instead of 60 miles per hour?

Explanation / Answer

A)

At speed of 60 miles per hour, the diesel truck will give average of 370/60 = 37/6 miles per gallon

Time taken for journey = 200/60 hours = 10/3 hours

Total cost = Cost of diesel + Cost of driver

=> 4.5 * (1200/37) + 22 * 10/3

=> 219.27$

b)

Let the constant speed be x

Time taken = 200/x

Total Cost = Cost of diesel + Cost of driver

=> 4,5 * (200/(370/x)) + 22 * 200/x

=> 900x/370 + 4400/x

Taking the derivative we get

900/370 - 4400/x^2 = 0

x = sqrt(4400 * 370/900) = 42.53

Hence the best constant speed is 42.53 mph

C) Cost with speed of 42.53 will be

=> 900 * 42.53/370 + 4400/42.53 = 206.90

Money saved = 219.27 - 206.90 = 12.37$

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