An offshore oil well is located in the ocean at a point which is 5 miles from the closest shore...

Question:

An offshore oil well is located in the ocean at a point which is 5 miles from the closest shore point on a straight shoreline. The oil is to be piped to a shore point that is 8 miles down the shoreline by piping it on a straight line underwater from the well to some shore point and then to the storage tank 8 miles down the shoreline.

a) If the cost of laying pipe is $100,000 per mile underwater and $75,000 per miles overland, where should the pipe be located to minimize the cost of laying the pipe?

b) What is the minimizing cost( to the nearest dollar)?

Cost of minimization

It is one of the tool used to find what number of labor N finally produces the resultant output at the minimum cost.

It can also be defined as the most effective method of sending goods while maintaining quality upto a certain level.

Answer and Explanation:

{eq}Cost\ of\ pipe=cost\ of\ under\ water\ +cost\ of\ load\\ =100000x+75000y\\ Now,\ substitue\ the\ value\ of x\\\\ C(y)=100000(\sqrt{25+(8-y)^2})+75000y\\ Now\ differentiating\ it\ with\ respect\ to\ y\\ C'(y)=100000(\frac{1}{2}(25+(8-y)^2)^\frac{-1}{2})+75000\\ =-100000(8-y)(25+(8-y^2))^{\frac{-1}{2}}+75000\\ Now,\ for\ critical\ points\\ C'(y)=0\\ \therefore y=2.33\\ to\ place\ is\ 2.33m\ to\ left\ of\ B.\\ Now,\\ x=\sqrt{25+(8-2.33)^2}=7.58\\ \therefore Cost\ minimum\ =100000*7.58+75000*2.33\\ =9.32750 {/eq}


Learn more about this topic:

Loading...
Data Mining: Identifying Functions From Derivative Graphs

from Math 104: Calculus

Chapter 10 / Lesson 8
8.4K

Related to this Question

Explore our homework questions and answers library