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)

1) the stages a new product goes through from beginning to end, i.e., the stages that a product passes through from introduction through growth, maturity, and decline. 2) the time from initial research and development to the time at which sales and support of the product to customers are withdrawn. 3) the period of time during which a product can be produced and marketed profitably.

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