posted by Katie .
Power Plant on the River Problem
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 |
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?
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.