An oil drilling rig located 14 miles off of a straight coastline is to be connected by a pipeline to a refinery 10 miles down the coast from the point directly opposite the drilling rig. Use calculus to solve and justify your answers to the questions below. Round mileage values to at least 4 decimal places. Round currency values to 2 decimal places. Be sure to justify your answers.
(a) Assume that laying underwater pipe costs twice as much as laying pipe on land. What path should the pipe take in order to minimize the total cost of the pipeline? What is the minimum cost if pipe costs $50,000 per mile on land?
(b) Suppose that underwater pipe costs 1.2 times as much as land pipe. What path should the pipe take? Hint: reconsider your picture.
(c) Suppose that underwater pipe costs α times as much as pipe on land (α > 1). Which combination of underwater and land-based pipe will minimize the total cost of the pipeline now? (Write the answer in terms of α.)
(d) How does the ideal amount of pipe under water and on land change as α increases? Does this seem to make sense to you?
(e) Determine the smallest possible value of α that makes sense for the problem setup (i.e. so the scenario in (b) doesn't occur).