optimization question Calc 1

Question

An island (I) is 3.5 mi from the nearest point (S) on a straight shoreline; that point (S)

is 8 mi from the power station (P). A utility company wants to lay electrical cable underwater

from the island to the shore at Q and then underground along the shore to the power station P.

The cost to lay underwater cable is $2400/mi and to lay underground is $1200/mi. Find the small

angle  shown in the figure below that minimizes the cost.

The equation should be set up in terms of the angle θ and then solved.

Answer 1

If we take a GUESS that the sketch looks like this:

Then:

Cost = 2400(distance underwater) + 1200(distance underground)

C = 2400d_W + 1200d_G

But d_W can be calculated by using sin(θ):

sin(θ) = 3.5/d_W or d_W = 3.5/sin(θ)

And, we can calculate "x" by using tan(θ):

tan(θ) = 3.5/x therefore x = 3.5/tan(θ) and therefore d_G = 8 - x = 8 - 3.5/tan(θ)

So C(θ) = 2400(3.5/sin(θ)) + 1200(8 - 3.5/tan(θ))

C(θ) = 8400/sin(θ) + 9600 - 4200/tan(θ)

C(θ) = 8400(sin(θ))^-1 + 9600 - 4200cot(θ)

Taking the derivative:

C'(θ) = -8400(sin(θ))^-2•cos(θ) + 4200csc²(θ)

C'(θ) = -8400cos(θ)/sin²(θ) + 4200/sin²(θ)

C'(θ) = [4200 - 8400cos(θ)]/sin²(θ)

Setting equal to zero to minimize:

[4200 - 8400cos(θ)]/sin²(θ) = 0 is only true if 4200 - 8400cos(θ) = 0

4200 - 8400cos(θ) = 0

4200(1 - 2cos(θ)) = 0

1 - 2cos(θ) = 0

2cos(θ) = 1

cos(θ) = 1/2

θ = 60°

Answer 2

First you have to draw the shore line and locate the island and the power station. It appears that you need to assume that the power station is located right on the shore line. In your question, the angle theta seems to be the angle between the underwater cable line connecting the island to the shore and the shortest line from the island to the shore (3.5 mi long and intersects the shore at 90 degree angle). These two lines and the shoreline together form a right-angle triangle. Express the underwater line length and the underground line length in terms of theta. Write the cost function in terms of theta and find its first derivative. Then find theta. I hope you got the idea.