Atruck is to be driven 300 miles on a freeway at a constant speed of x miles per hour. speed laws require that 35 ≤ x ≤ 55. assume that fuel costs $1.35 per gallon and is consumed at the rate of 2 + 1 600 %01 x 2 galloons per hour. given that the driver’s wages are $13 per hour, at what speed should the truck be driven to minimize the truck owner’s expenses?

x = 55 Miles per hour

Explanation:

Length of trip, L = 300 miles

If "x" is the velocity in miles per hour, time t taken to complete the trip  = Length / velocity  

t = L / x

There are two components of cost:

The cost per hour for fuel is C dollars, and the driver is paid W dollars per hour.

Hence the cost for the entire trip = The cost per hour for fuel x total number of hours + Hourly cost driver x total number of hours = C*t + W*t

C = 2 + (1/600)*x²,    W = 13.00   and    t = 300 / x

Hence our total cost of the trip,  

TC(x) = (2 + (1/600)*x²)*(300 / x) + 13*(300 / x)  

TC(x) = (4500 / x) + (x / 2)

In order to minimise this total cost, we need to differentiate this with respect to x and equate it to zero:

TC’(x) = ((4500 / x) + (x / 2))’ = 0

(- 4500 / x²) + (1 / 2) = 0

this equation gives x₁ = - 94.868 and x₂ = 94.868

as we know that  35 ≤ x ≤ 55

we can say that x = 55 Miles per hour (which is the maximum value)

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