Problem:

A) A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side, 3,000 meters downstream.The cost of running the cable under the water is $5 per meter, while the cost over land is $4 per meter.What is the most economical route over which to run the cable and what will the minimum cost be? Include both the lengths of the cable needed under water and over land.

B) For the summer, the company installing the cable in the problem above has hired Frank Kornercutter as a consultant.Frank, recalling a problem from his calculus course in college, asserts that no matter how far downstream the factory is located (beyond 1,200 meters), it would be most economical to have the cable reach the opposite bank 1,200 meters downstream from the power plant.The supervisor, amused by Frank's naivet, replies, "Any fool can see that if the factory is farther away, the cable should reach the opposite bank farther downstream.It's just common sense!"Of course, Frank is no common fool, but is he right?Why?Justify your answer using calculus.