Question

A truck gets 377/x mpg when driven at a constant speed of x mph (between 25 and 75 mph). If the price of fuel is $1 per gallon and the driver is paid $8 per hour, at what speed between 25 and 75 mph is it most economical to drive? (Give your answer correct to the nearest full mph)
I assume "most economical" means lowest cost per mile.

The driver's wages:

$/mi = $/hr * hr/mi = $/hr / (mi/hr)

The truck:

$/mi = $/gal * gal/mi = ($/gal) / (mi/gal)

If speed is x, and y is cost/hr, including the driver's wages

y = 8/x + 1 * 1/(377/x)
y = 8/x + x/377

For minimum cost/mile,

y' = -8/x^2 + 1/377
x^2 = 3016
x = 54.9

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