At what speed between 50 and 70 mph will it be most economical to drive?

Question

a truck achieves 600 / x mpg when driven at a constant speed of x mph (between 50 and 70 mph). If the price of fuel is $ 3 per gallon and the driver is paid $ 18 / h. At what speed between 50 and 70 mph will it be most economical to drive?

Answer 1

So, we have the function f(x) = 600/x for the miles per gallon used by the truck. In order to find the most economical speed, we need f(x) to be maximized (which will maximize the mpg of the truck).

To find local maximums (or minimums) of a function, we must find the derivative, and set it equal to 0. To find the derivative, first rewrite f(x). Remember that 1/x = x^-1:

f(x) = 600/x = 600 x^-1

Now, use the power rule:

f'(x) = 600 (-1x^-2) = -600/x²

If we set f'(x) equal to 0, we get an equation that we can't solve. (Try multiplying both sides by x², and you'll end up with -600 = 0, which isn't helpful).

What this tells us is that there are no local maximums (or minimums) for this function. So, the only way to find the maximum is to check the endpoints that were given:

f(50) = 600/50 = 12

f(70) = 600/70 ≈ 8.6

Since f(50) > f(70), the most economical speed is 50 miles per hour!