Baltrum

A small resort is situated on an island off of a part of the coast of mexico that has a perfectly straight?

north-south shoreline. The point P on the shoreline that is closest to the island is exactly 4 miles from the island. Ten miles south of P is the closest source of freshwater to the island. A pipeline is to be built from the island to the source of fresh water by laying pipe underwater in a straight line form the island to a point Q on the shoreline between P and the water source, and then laying ipe on land along the shoreline form Q to the source. It costs 1.7 times as much money to ,ay pipe in the water as it does on land. How far south of P should Q be located in order to minimize the total construction costs? Hint: You can do this problem by assuming that it costs one dollar per mile to lay pipe on land, and 1.7 dollars per mile to lay pipe in the water. You need to minimize the cost over the interval [0,10] of the possible distances from P to Q

Public Comments

1. Let x be the distance The cost of the pipeline is (10-x) + 1.7*sqrt(16+x^2). To find the minimum cost we take the derivate with respect to x and set it equal to 0 to get: -1 +1.7(2x)(1/2)/sqrt(16+x^2) = 0 1 = 1.7x/sqrt(16+x^2) sqrt(16+x^2) = 1.7x square both sides to get: 16 + x^2 = 2.89x^2 16 = 1.89x^2 So x = sqrt(16/1.89)
2. Let the distance from P to Q be x. Then the distance from Q to the water source is (10-x). The distance from the island to Q is the hypotenuse of a right triangle with sides 4 and x. This distance is: sqrt(4^2 + x^2) sqrt = square root The cost is thus given by: C = 1*(10-x) + (1.7)*sqrt(x^2+16) = 10 - x + (1.7)*sqrt(x^2+16) To minimize C, we find the derivative of C, and find where it is equal to 0. dC/dx = -1 + (1.7) * (1/2) * (x^2+16)^(-1/2) * (2x) Setting this equal to 0 and solving for x, we get x equal to approximately 2.91. So to minimize cost, the distance from P to Q should be approximately 2.91.