Expert: Paul Klarreich - 7/26/2009

Question
A small business uses a minivan to make deliveries. The cost per hour for fuel is F= (v^2) / 360, where 'v' is the speed of the minivan (in mph). The driver is paid $10/hour. Find the speed that minimizes the cost of a 110-mile trip.

I know that by taking the first derivative and finding the critical value(s), I can get the value that gives me a minimum, but I do not know how to work the wage and distance into the problem. I appreciate any feedback.
Thank you for your time,
Robert

Answer
Questioner: Robert
Country: United States
Category: Calculus
Private: No
Subject: Minimum Cost Problem
Question: A small business uses a minivan to make deliveries. The cost per hour for fuel is F= (v^2) / 360, where 'v' is the speed of the minivan (in mph). The driver is paid $10/hour. Find the speed that minimizes the cost of a 110-mile trip.

I know that by taking the first derivative and finding the critical value(s), I can get the value that gives me a minimum, but I do not know how to work the wage and distance into the problem. I appreciate any feedback.
Thank you for your time,
Robert
.............................
Cost = fuel cost + labor cost.

labor cost = 10t, where t is the elapsed time.

fuel cost = (v^2 /360) t

distance = rate * time

110 = vt

t = 110/v

Cost = (v^2 /360) t + 10 t

Cost = (v^2 /360)(110/v) + 10 (110/v)

Cost = 110 v /360 + 1100/v

Cost = 11 v /36 + 1100/v

C' = 11/36 - 1100/v^2

11/36 - 1100/v^2 = 0

11/36 = 1100/v^2

11 v^2 = 36 * 1100

v^2 = 3600

v = 60