A company uses a truck to deliver the products. To estimate costs, the manager m

Question

A company uses a truck to deliver the products. To estimate costs, the manager models the gas consumption by the function G(x)=1/250(1,200/x+x) gal/mile, assuming the truck is driven by a constant speed of x miles per hour, for x ge 5. The driver is paid $20 per hour to drive the truck 250 miles, and gasoline costs $1.90 per gallon. Find an expression for the total cost C(x) of the trip, and use C(x) to determine if the costs is increasing or decreasing when the truck is driven 40 miles per hour, 50 miles per hour and 60 miles per hour. Interpret the results.

Explanation / Answer

The two parts of the cost C(x) are the price of gas and the price of the driver.

The price of fuel is G(x)(trip length)(dollars per gallon)

The price of the driver is [(trip length)/(driving speed, x)](dollars per hour)

C(x) = [(1/250)[(1200/x) + x]gallons per mile](1.90 dollars per gallon)(250 miles) +
[(250 miles)/(x miles per hour)](20 dollars per hour)

C(x) = 1.90[{1200/x) + x] + 5000/x

Differentiate cost with respect to driving speed and investigate for speeds of 40, 50, 60 miles per hour.

C(x) = 2280/x + 1.90x + 5000/x = 7280/x + 1.90x

C'(x) = dC(x)/dx = - 7280/x2 + 1.90

C'(40) = - 7280/402 + 1.90 = - 2.65 dollars per mile

C'(50) = - 7280/502 + 1.90 = - 1.01 dollars per mile

C'(60) = - 7280/602 + 1.90 = - 0.12

Transport costs per mile are dropping as speed increases.

Optimal speed is x for C'(x) = 0

- 7280/x2 + 1.90 = 0

7280/1.90 = x2

x = (7280/1.90) = ± 62 miles per hour.

Negative speed could apply to velocity in the return direction. If the truck were empty, however, its gas consumption would be different and G(x) would be different. The optimal transport cost is realized using "back-hauling", the practice of dropping off a load at the destination and picking up another before returning home.

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