Expert: Paul Klarreich - 7/10/2007

Question
hi i have this optimization problem that i'm having a lot of problem with and would love help with! its for finding minimum cost:

the cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per house when cruising at 10 miles per hour. Apart form fuel, the cost of running this ferry (labor, maintenance and so on) is $675 per hour. At what speed should it travel so as to minimize the cost PER MILE traveled?

thanks!

Answer
Questioner:   kaitlin
Category:  Calculus

Question:  hi i have this optimization problem that i'm having a lot of problem with and would love help with! its for finding minimum cost:

the cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per house

>> per house?  What is a house?  Are you sure you didn't mean to type 'per hour?'  I think that makes sense unless you define what a 'house' is.


when cruising at 10 miles per hour. Apart form fuel, the cost of running this ferry (labor,maintenance and so on) is $675 per hour. At what speed should it travel so as to minimize the cost PER MILE traveled?

thanks!


Let:
  v = the velocity of the boat.
  F = cost of fuel per mile.

At 10 mph, if we use $100/hour of fuel, that will be $10/mile.

If cost is proportional to the cube of the speed, we write that as:

F = kv^3  

and we use the known fact that when  v = 10, F = $10/mile.

10 = k (10)^3 = 1000k

so  k = 1/100

Now to the optimization part:

Let C = total cost of running the ferry, per mile.  That has two components:

Labor:  That costs $675 per hour.  But if we travel at  v mph, that cost is  675/v.

Fuel:  F = v^3/100

That totals:

C = 675/v + v^3/100

Now we are ready to go.  Do our standard optimization stuff:

C' = - 675/v^2 + 3v^2/100

Set that equal to zero and solve:

- 675    3v^2
----- + ----- = 0
 v^2    100

 675    3v^2
----- =  -----
 v^2    100

67500 = 3v^4

22500 = v^4

v^2 = sqrt(22500) = 150

v = sqrt(150) = 5 sqrt(6) = 12.247448713915890490986420373529 mph