$2$

OPTIMIZATION PROJECT

MTH $1310$

SPENCER$-$TYREE

OPTIMIZATION PROJECT

An oil company wants to build a pipeline to take oil from an oil well to a refinery. Unfortunately, the well and the refinery are on either side of a straight river which is $10$ miles wide, and they are $60$ miles apart along the coastline $($that is$,$ if you want to go from the well to the refinery you must first cross $10$ miles of river and then go $60$ miles along the side of the river$).$ The company has hired you to figure out the cheapest way to build the pipeline, and you need to clearly explain your solution so that the less mathematically$-$sophisticated oil people will understand.

It costs $$250$ per mile to lay pipe across the river but only $$180$ per mile to lay the pipe over land. There is also one other potential cost: It costs extra money each time that you have a bend in the pipeline. If you go straight across the river and use an L$-$shaped bend it costs an extra $$200.$ If you lay the pipe diagonally across the river but hit land before you get to the refinery, you have to use a slanted L$-$shaped bend $($like an obtuse angle$).$ These have to be custom made and they cost $$1505.$ If you go directly diagonally across the river to the refinery without touching 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 doing the following:

$($a$)$ Draw a diagram of the situation. Include any variables that you are going to use in the rest of your answer.

$($b$)$ Calculate the cost of building the pipeline for a few different situations:

$($i$)$ How much would it cost to build the pipeline $10$ miles straight across the river, then make a $90$ degree bend and go $60$ miles along the side of the river?

$($ii$)$ How much would it cost to go diagonally across the river, without going on land at all?

$($iii$)$ Suppose $P$ is the point directly across the river from the oil well. How much would it cost to go diagonally across the river to a point $10$ miles along the bank from $P,$ then bend the pipeline and go the remaining $50$ miles along the side of the river?

$($iv$)$ How much would it cost to go diagonally across the river to a point $50$ miles along the bank from $P,$ then bend the pipeline and go the remaining distance along the side of the river?

Exercise $2.$ Calculate the cheapest way to build the pipeline for the situation given above by finding the minimum cost among ALL possible locations for $P.$

$($i$)$ You will need to write a general expression for the cost of building the pipeline ${?}^1,$ and then use calculus to minimize the cost.

$($ii$)$ Use calculus $($such as the $1$st derivative test$)$ to prove that your result is a minimum.

$($iii$)$ What is the effect of the extra cost for a bend in the pipeline? If there was no extra cost for the bend, would you have a different answer for what the cheapest path would be$?$

Present your results in a written letter$/$report$.$ This needs to be typed, written grammatically correctly, and written in an organized manner. The report only needs to clearly summarize your findings$/$costs of the different options and present the recommendation for what you think the company should do if they are trying to save money Please attach all of your work behind the letter.

Timeline: This project is due Wednesday, April $10$th by $11$:$59$pm