Quote:
Originally Posted by superhume
What would be the approach for this problem:
A power house, P, is on one bank of a straight river 200m wide, and a factory, F, is on the other bank 400m downstream from P. The cable has to be taken across the river under water at a cost of $12.00/m. On land the cost is $6.00/m. What path should be chosen so the cost is minimized?
Any help would be much appreciated.

W = distance under water
L = distance on land
Cost = 12*W + 6*L
If the cable is run underwater in a straight line to the opposite bank to some point between the factory and a point straight across from the power house, then the distance underwater is the hypotenuse of the right triangle with legs 200m and 400L meters.
W^2 = 200^2 + (400L)^2
W = sqrt(40000 + 160000  800*L + L^2)
W = sqrt(200000  800*L + L^2)
Now plug W back into the cost equation.
Cost = 12*sqrt(200000  800*L + L^2) + 6*L
One way to find the minimum cost is to take the derivative of cost set it equal to 0, and solve for L.
d/dL(cost) = (6*sqrt(200000  800*L + L^2) + 12*L  4800)/sqrt(200000  800*L + L^2)
Setting the der of cost to 0 gives:
0 = 6*sqrt(200000  800*L + L^2) + 12*L  4800
(4800  12*L)/6 = sqrt(200000  800*L + L^2)
Squaring both sides gives:
(144*L^2  115200*L + 23040000)/36 = L^2  800*L + 200000
36*L^2  28800*L + 7200000 = 144*L^2  115200*L + 23040000
0 = 108*L^2  86400*L + 15840000
Using quadratic formula gives L = 284.53 or L = 515.47
Throw away the value greater than 400 and plug the value for L back into the W question to solve for W.
W = sqrt(40000 + (400  284.53)^2) = 230.94
Min cost = $4478.46. The cable should be run to a point 284.53m from the factory on the opposite bank of the river.