An island is 4 miles offshore in a large bay A water pipelin

An island is 4 miles offshore in a large bay. A water pipeline is to be run from a water tank on the shore to the island, as indicated in the figure. The pipeline costs $40,000 per mile in the ocean and $20,000 per mile on the land. Compute the cast for the angles below (include drawing): Express the total cost of the pipeline in terms of 0. Your answer should look like Cost = a - btan(theta) + csec(theta). How many miles of pipe (to two decimal places) should be laid on land and how many mile\'s placed in the water for the total cost to be minimum? What is the cost of the shortest pipeline from the island to the water tank? Explain why the shortest pipeline from the island to the water tank is not the least costly. (I\'m curious: What is theta for this situation?)

Solution

length of pipe offshore=4 sec length of pipe on shore =10-4 tan X total cost=( 40,000X 4sec)+(20000X(10-4tan)) cos=4/X X=4 sec costheta 1 -0.76 0.15 0.53 -0.95 tantheta 0 -0.86 -6.41 1.62 0.32 sec theta 1 -1.32 6.48 1.9 -1.05 angle 0 15 30 45 60 4sectheta 4 -5.27 25.9 7.61 -4.2 4tan theta 0 -3.42 -25.6 6.479 1.28 ocean cost 160000 -210613 1037267 304575.1 -167994 shore cost 200000 268479.5 712426.5 70417.98 174396.8 cost 360000 57866.67 1749694 374993.1 6402.418