An electrical powergenerating plant is located on the shores

An electrical power-generating plant is located on the shores of a river at point E. Point P lies directly across on the opposite shore at which point the river is W km wide. A large factory is under construction at point F which is located L km from P. See diagram

Solution

The cost of laying the cable under water is A dollars/Km and on land is B dollars/Km

Now, for a distance, x, the length of the cable that needs to be laid under water = sqrt(w^2 + x^2)

Cost of laying under water = A*sqrt(w^2 + x^2)

Cost of laying on land = B*(L-x)

Total cost = C = A*sqrt(w^2 + x^2) + B*(L-x)

Now for optimal position of Q, we differentiate the cost with respect to x

dC/dx = Ax/ sqrt(w^2 + x^2) - B

This has to be zero for minimum value of C, that is

Ax/ sqrt(w^2 + x^2) = B; That is, A^2 * x^2 = B^2 * (w^2 + x^2)

Hence X^2 = (B^2 * w^2) / (A^2 - B^2)

That is x = sqrt [(B^2 * w^2) / (A^2 - B^2)]