A power house, P, is on one bank of a straight river 200 m wide, and a factory, F, is on the opposite bank 400 m down stream from P. The cable has to be taken across the river under water at a cost of $6/m. On land the cost is $3/m. What path should be chosen so that the cost is minimized?
Call the point across the river from the power house P' Call the point of intersection of the cable with the opposite bank C, which defines right triangle PP'C Let x be the distance from P' to C The two legs of triangle PP'C are 200 and x So the hypotenuse (the distance the cable travels underwater) is sqrt(200^2 + x^2) The distance the cable travels along the shore is (400 - x) (the distance from C to F) cost under water: 6sqrt(200^2 + x^2) cost on land: 3(400 - x) 1200 - 3x Total cost 6sqrt(200^2 + x^2) + 1200 - 3x Find derivative of cost: 6(2x) / (2sqrt(200^2 + x^2)) - 3 6x/sqrt(200^2 + x^2) - 3 Where derivative 0 will be a min for the cost 6x / sqrt(200^2 + x^2) 3 2x sqrt(200^2 + x^2) 4x^2 200^2 + x^2 3x^2 200^2 x^2 200^2 / 3 x +/- sqrt(200^2 / 3) +/- 200 / sqrt(3) but discard the negative root (since this won't be the minimum cost), so the distance from P' to the cable hitting land should be 200 / sqrt(3) 200 / (3^(1/2))
