An offshore oil well must be connected by pipe to the refinery. The oil-well is 4 km off-shore. The refinery is on the coast, 9 km from the nearest point of land to the oil well. It costs $ per kilometre to lay pipe along the shore, but 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.
(a)
Give the total cost as a function of the distance between the point directly opposite the oil well and the place where the pipe comes ashore. This should be a formula involving , and .
(b)
Find the -value of the critical point of this cost function. This point will be a function of .U
(c)
For which value of is U the critical point at the refinery, x=9 ?
(d)
When U is twice this value, what is the x-value of the critical point? Answer with a decimal accurate to 0.1 km

Question

An offshore oil well must be connected by pipe to the refinery. The oil-well is 4 km off-shore. The refinery is on the coast, 9 km from the nearest point of land to the oil well. It costs $  per kilometre to lay pipe along the shore, but  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.
(a)	
Give the total cost as a function of the distance  between the point directly opposite the oil well and the place where the pipe comes ashore. This should be a formula involving  ,  and  .
(b)
Find the -value of the critical point of this cost function. This point will be a function of .U
(c)
For which value of is U the critical point at the refinery, x=9 ?
(d)
When U is twice this value, what is the x-value of the critical point? Answer with a decimal accurate to 0.1 km