Optimization word prob. rate speed cost: cost of fuel to propel ship

[supatramp]

[FONT=&quot]Hi there, have a question to do with a calculus word problem.

The cost of fuel to propel a particular ship through the water is $(v/11)^2 per hour, where v is the ship's speed in nautical miles per hour. Other costs incurred are $4 per hour, regardless of speed. At what speed should the ship travel to minimise its cost per nautical mile travelled?

I can get the total cost per hour at v^2/121 + 4, then using the distance= rate x time formula, dividing the equation by v, which I am pretty sure I want to solve for, = v/121 + 4/v. Do I just take the derivative of this and set it to zero, or start all over again, 'cause I have been looking at it so long I am a little cross eyed???[/FONT]

 

[stapel]

Hi there, have a question to do with a calculus word problem.

The cost of fuel to propel a particular ship through the water is $(v/11)^2 per hour, where v is the ship's speed in nautical miles per hour. Other costs incurred are $4 per hour, regardless of speed. At what speed should the ship travel to minimise its cost per nautical mile travelled?

I can get the total cost per hour at v^2/121 + 4, then using the distance= rate x time formula, dividing the equation by v, which I am pretty sure I want to solve for, = v/121 + 4/v. Do I just take the derivative of this and set it to zero, or start all over again, 'cause I have been looking at it so long I am a little cross eyed???

I think your reasoning was something like this:

. . .cost per unit time t: C_t = (v^2/121) + 4

. . .total cost for total time: C = C_t * t

. . .cost per unit distance d: C_d

. . .total cost for total distance: C_d * d

. . .uniform-rate equation: d = rt

Then, assuming the boat never changes speed, we have:

. . .total cost for total distance: C_d * rt

The total cost being the total cost, we can equate and get:

. . .C_d * rt = C_t * t

Assuming t is not zero, we can divide through:

. . .C_d * r = C_t

Assuming the rate is not zero, we can divide through:

. . .C_d = C_t / r

So your "cost per unit distance" equation seems valid:

. . .C_d = (v^2 / 121 + 4) / v = v/121 + 4/v

Now differentiate, etc, and see if you get an answer that seems reasonable. (For instance, if you get a negative value, suggesting that the ship should always be in "reverse" gear, probably something went wrong somewhere.) ;-)