An offshore oil well must be connected by pipe to the refinery. The oil-well is a distance d= 5 km off-shore.

Question

An offshore oil well must be connected by pipe to the refinery. The oil-well is a distance d= 5 km off-shore. The refinery is on the coast, 10 km from the nearest point of land to the oil well (labelled P in the diagram). The coastline is perfectly straight. It costs $S per kilometre to lay pipe along the shore, but U times as much per km to lay pipe underwater. The pipe may be laid either straight to the refinery, or to an intermediate point on the coast then along the coast as shown in the diagram.

(a) Give the total cost as a function of the distance x between the point P directly opposite the oil well and place A where the pipe comes ashore. This should be a formula involving x, S and U

(b) Find the x-value of the critical point of this cost function. This point will be a function of U

(c) For which value of U is the critical point at the refinery, x=10?

(d) When U is twice the value in (c), what is the x-value of the critical point? Answer with a decimal accurate to 0.1 km.

Answer 1