An oil company wants to build a pipeline to take oil from an oilwell to a refinery. Unfortunately, the well and the refinery are oneither side of a straight river which is 10 miles wide, and theyare 50 miles apart along the coastline (that is, if you want to gofrom the well to the refinery you must first cross 10 miles ofriver and then go 50 miles along the side of the river). Thecompany has hired you to figure out the cheapest way to build thepipeline, and you need to clearly explain your solution so that theless mathematically-sophisticated oil people will understand.
It costs $200 per mile to lay pipe across the river but only$160 per mile to lay the pipe over land. There is also one otherpotential cost: It costs extra money each time that you have a bendin the pipeline. If you go straight across the river and use anL-shaped bend it costs an extra $150. If you lay the pipediagonally across the river but hit land before you get to therefinery, you have to use a slanted L-shaped bend (like an obtuseangle). These have to be custom made and they cost $975. If you godirectly diagonally across the river to the refinery withouttouching any land, then you do not have to pay extra for a bend(since there will not be one).
Exercise 1. Find the cheapest path to lay the pipeline by doingthe following:
(a) Draw a diagram of the situation. Include any variables thatyou are going to use in the rest of your answer.
(b) Calculate the cost of building the pipeline for a fewdifferent situations: (i) How much would it cost to build thepipeline 10 miles straight across the river, then make a 90 degreebend and go 50 miles along the side of the river? (ii) How muchwould it cost to go diagonally across the river, without going onland at all? (iii) Suppose P is the point directly across the riverfrom the oil well. How much would it cost to go diagonally acrossthe river to a point 10 miles along the bank from P, then bend thepipeline and go the remaining 40 miles along the side of the river?(iv) How much would it cost to go diagonally across the river to apoint 40 miles along the bank from P, then bend the pipeline and gothe remaining distance along the side of the river?
(c) Calculate the cheapest way to build the pipeline for thesituation given above by finding the minimum cost among ALLpossible locations for P. (i) You will need to write a generalexpression for the cost of building the pipeline1 , and then usecalculus to minimize the cost. (ii) Use either the first or secondderivative test to prove that your result is a minimum. (d) What isthe effect of the extra cost for a bend in the pipeline? If therewas no extra cost for the bend, would you have a different answerfor what the cheapest path would be?
