Power Plant on the River Problem

Question

As the construction manager for the Lightem Up Power Company , you make decisions to hep minimize costs of the company's construction projects. Today you've been asked to determine the path of a pipe that will house communication lines from the factory to the power plant pictured here.

Factory . F_______7 miles_______
          |      river           |
4                           miles                                          
          |______________________|
                                .P Power
                                   Plant

You must connect a pipe from point F at the factory to point P at the power plant. It will cost $12 per foot to lay the pipe on land and $21 per foot to lay the pipe in the water. Where should the pipe be placed so as to minimize the total cost of laying the pipe?

Reiny · Answer

let the point directly across from P (the power plant) be Q.
then FQ = 7 miles

let R be a point between F and Q so that RQ = x miles, and FR = 7-x miles.
RP^2 = x^2 + 16
RP = (x^2+16)^(1/2)

So the path of the pipeline is PR + RF.
Cost of pipe on land: $12 per foot = $12(5280) per mile
= $63360 per mile
Cost of pipe in water: $21 per foot or
$110880 per mile

Cost = 110880*RP + 63360*FR
= 110880(x^2+16)^)1/2) + 63360(7-x)

d(Cost)/dx = (1/2)(2x)(110880)(x^2+16)^(-1/2) - 63360x
= 0 for a minimus of Cost
this reduced to

7x/√(x^2+16) = 4 (I divided by 15840x)
cross-multiplying and squaring both sides gave me
49x^2 = 16x^2 + 256
33x^2 = 256
x = 16/√33 = 2.785

then 7-x = 4.215

They should aim for a point 4.215 miles from the factory, then cross the river to the power plant.