# Optimation Problem

• Jun 21st 2010, 07:08 AM
praharsh14
Optimation Problem
A house (H) is located on an island that is 200 north of a point (A) on the shore. A Power station (p) is located on the shoreline at a point 400 m west of Point A. An electric cable is going to be laid from the power station to the house. It costs $12/m to lay underwater and$6/m to lay underground. How should the cable be placed so that the cost is a minimum and what is this minimum cost?

• Jun 21st 2010, 07:11 AM
Ackbeet
What have you done so far?

Clarification question: is that $12/m or$12/mm for the underwater cost?
• Jun 21st 2010, 07:12 AM
praharsh14
I cant seem to make a cost equation
its $12/meter Thanks • Jun 21st 2010, 07:15 AM Ackbeet Well, to start you out, I'd go with$\displaystyle \text{cost}=\$12d_{w}+\$6d_{g}$, where$\displaystyle d_{w}$is the length of underwater cable, and$\displaystyle d_{g}$the length of underground cable. So, I'm assuming you're going to want to go some distance east from the power station, underground, and then at some point you're going to go underwater. Is this correct? • Jun 21st 2010, 07:18 AM praharsh14 Thats looks correct and could you please elaborate on the equation, on how I could make that a derivative • Jun 21st 2010, 07:20 AM Ackbeet Ok. You need to find$\displaystyle d_{g}$and$\displaystyle d_{w}\$ in terms of one parameter: the point at which the cable leaves the shore. Draw a picture and start labeling things. What do you get?
• Jun 21st 2010, 07:36 AM
SpringFan25
You description is pretty ambiguos. The interpretation i ahve used is in the attachment.

Attachment 17949

There are only 2 routes the cable is likely to take (p ->h), (p -> a -> h). Try them both and see which is cheaper.
• Jun 21st 2010, 07:38 AM
praharsh14
Shouldn't it be that the cable goes a certain distance underground, and then underwater instead of going all the way to A first
• Jun 21st 2010, 10:38 AM
HallsofIvy
Quote:

Originally Posted by SpringFan25
You description is pretty ambiguos. The interpretation i ahve used is in the attachment.

Attachment 17949

There are only 2 routes the cable is likely to take (p ->h), (p -> a -> h). Try them both and see which is cheaper.

I like your disclaimer! And this is not accurate- there are an infinite number of possible routes and neither of the ones you give is likely to be optimal. Put the cable from H to some point X between H and A. Let "x" be the distance from H to X. The cost of laying that cable is 6x.

The distance remaining from X to A is 200- x and the distance across from A to P is 400 so you can use the Pythagorean theorem to find the straight line distance form X to P. Multiply that by 12 to find the cost of laying that cable.

The total cost is the sum of those two costs. Differentiate that sum with respect to x and set the derivative equal to 0 to find the x that gives optimal cost.

By the way, the correct word is "optimization", not "optimation".