Question

a power station needs to supply power lines to a location 10 miles downstream and on the other side of a river that is 0.75 miles wide. Power lines cost $7 per foot on land and $10 per foot under water. What is the most economical path? What is the minimum cost?

I know I have to find the cost function first and then take the derivative, make it equals to zero and get x as the minimum cost, but I still need help in detail explanation please.

Answer 1

Of course you made a sketch.

Let the route through the water end up at a point P
Let the point directly opposite be A and the station as B
Let AP = x, then PB = 10-x

let y be the distance in the water, then
y^2 = x^2 + .75^2
x = (x^2 + .5625)^(1/2)

cost = 10(x^2 + .5625)^(1/2) + 7(10-x)
d(cost)/dx = 5(x^2 + .5625)^(-1/2) (2x) - 7
= 10x/√(x^2 + .5625) - 7 = 0 for a max/min

10x/√(x^2 + .5625) = 7
square both sides,
100x^2/(x^2 + .5625) = 49
100x^2 = 49x^2 + 27.5625
51x^4 = 27.5625
x = ....

take over, but make sure to check my calculations