# Optimization problem - minimize cost

Optimization problem -- minimize cost

## Homework Statement

A motor is floating on a buoy in a large pool with a perfectly flat shoreline. The motor is 3ft from the edge of the pool. A circuit to control operation of the motor is located 100ft from the closest point of the motor to the edge of the pool. Cable must be installed to control the motor. It costs 2.50/ft to run cable underwater, and 1.50/ft to run it on land. Find the proportions (amount of cable on land and amount on water) to minimize cost.

## Homework Equations

Cost= (2.50w)+(1.50L) trying to minimize this function.
where w= underwater cable length
L= land cable length

w^2 = (100-L)^2 + 3^2 This is the "top" triangle that is formed. 3 is a fixed length, one side of the triangle. 10-L is the other side, and w is the hypotenuse.

## The Attempt at a Solution

C=2.50[(100-L)^2+9]^.5 + 1.50L

Use chain rule to find C': (5/4)[(100-L)^2+9]^-.5(2L-200) + 3/2
Set equal to zero and solve for L....right
Rewriting this gives a rational function (because of the -1/2 power) plus a constant.

Is this correct thus far? I tried solving this by rewriting as a rational function. I know that I only need to concern myself with the numerator of rational functions to make the whole thing equal to zero, but I have that constant out there. I tried finding a common denominator, but then the only way I can make the function equal to zero is by setting length of land cable as zero.

Redbelly98
Staff Emeritus
Homework Helper

## The Attempt at a Solution

C=2.50[(100-L)^2+9]^.5 + 1.50L

Use chain rule to find C': (5/4)[(100-L)^2+9]^-.5(2L-200) + 3/2
Set equal to zero and solve for L....right
Rewriting this gives a rational function (because of the -1/2 power) plus a constant.

Is this correct thus far?

Yes, looks good so far.

Can you get the square-root term so it's on one side of the equation, all by itself? Then you'd just need to square everything to get rid of the radical.