Expert: Paul Klarreich - 1/31/2008

Question
I am having trouble figuring out this problem:

Assume that operating cost of a certain truck (excluding driver’s wages) is 12 + x/6 cents per mile when the truck travels at x mi/hr.  If the driver earns $6 per hour, what is the most economical speed to operate the truck on a 400-mi turnpike where the minimum speed is 40 mi/hr and the maximum speed is 70 mi/hr?

Thank you very much for you help in advance!

Answer
Questioner:   Jonathan
Category:  Calculus
Private:  No
 
Subject:  Most economical speed to operate (Max/min)
Question:  I am having trouble figuring out this problem:

Assume that operating cost of a certain truck (excluding driver’s wages) is 12 + x/6 cents per mile when the truck travels at x mi/hr.  If the driver earns $6 per hour, what is the most economical speed to operate the truck on a 400-mi turnpike where the minimum speed is 40 mi/hr and the maximum speed is 70 mi/hr?

Thank you very much for you help in advance!
.....................................
Hi, Jonathan,

I am going to assume that 'most economical speed' means 'lowest cost per mile.'  So you must express Total cost per mile in terms of x.

TCPM(x) = wages(x) + opcost(x)

If the driver gets $6 (600 cents) to drive at x mph, then wages(x) = 600/x, so:

TCPM(x) = 600/x + 12 + x/6

Now do your standard max-min stuff:

TCPM'(x) = -600/x^2 +1/6, set = 0:

-600/x^2 +1/6 = 0

600/x^2 = 1/6

x^2 = 3600

x = 60

That's your minimum, I think.