Math Help - Optimization, Minimization Calculus Problem--I'm Stuck!

1. Optimization, Minimization Calculus Problem--I'm Stuck!

The problem I'm working on is...

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$20/hr, at what speed is it most economical for the trucker to drive?

I have been searching for similar problems in my text book and class notes but am unable to find one. I'd greatly appreciate any help or similar examples!

Thanks!

2. Re: Optimization, Minimization Calculus Problem--I'm Stuck!

The trick of it is to find what you want to optimize. In this case it would be miles per dollar.

You are given miles/gallon, dollars/gallon, miles/hour, and dollars/hour.

Spoiler:
(miles/gallon) / (dollars/gallon) = fuel miles/dollar
(500/x) / (2.8) = 500/(2.8x)

miles/hour / dollars/hour = driver miles/dollar
x / 20 = x/20

fuel miles/dollar + driver miles/dollar = total miles/dollar
500/(2.8x) + x/20 = f(x)

Once your equation is set up it is straight forward optimization.