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

Answer :

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 = $\frac{1}{\frac{500}{x}}\times 2.80$

$=\frac{2.80x}{500}$ dollar per mile,

The driver earns $10 per hour.

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

$=\frac{10}{x}$ dollars per mile,

Thus, total cost,

$P = \frac{10}{x} + \frac{2.8x}{500}$

Differentiating with respect to x

$\frac{dP}{dx} = -\frac{10}{x^2} + \frac{2.8}{500}$

Again differentiating with respect to x,

$\frac{d^2P}{dx^2}=\frac{30}{x^3}$

When,

$\frac{dP}{dt} = 0$

$\implies 2.8 x^2 = 5000$

$\implies x^2 = 1785.71428$

$\implies x = 42.26$

At x = 42.26, $\frac{d^2P}{dx^2}$ = positive.

Hence, the speed would be 42.26 miles per hour.