Question

A truck gets 500/x miles per gallon (mpg) when driven at a constant speed of x mph, where 40 ≤ x ≤ 80. If the price of fuel is $2.80/gal and the driver is paid $10/hr, at what speed is it most economical for the trucker to drive? (Round your answer to two decimal places.)

Answer 1

42.26 mph (approx)

Explanation :

Given,

Fuel consumption = 500/x miles per gallon,

Speed = x miles per hour

Also, hourly cost = $2.80/gal,

So, fuel cost =
`(1)/((500)/(x))* 2.80`

`=(2.80x)/(500)` dollar per mile,

The driver earns $10 per hour.

So, labour cost = (10 dollars per hour) × (1/x hr per mile)

`=(10)/(x)` dollars per mile,

Thus, total cost,

`P = (10)/(x) + (2.8x)/(500)`

Differentiating with respect to x

`(dP)/(dx) = -(10)/(x^2) + (2.8)/(500)`

Again differentiating with respect to x,

`(d^2P)/(dx^2)=(30)/(x^3)`

When,

`(dP)/(dt) = 0`

`\implies 2.8 x^2 = 5000`

`\implies x^2 = 1785.71428`

`\implies x = 42.26`

At x = 42.26,
`(d^2P)/(dx^2)` = positive.

Hence, the speed would be 42.26 miles per hour.