An oil refinery is located on the north bank of a straight river that is 2 km wi
ID: 2828135 • Letter: A
Question
An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $400,000/km over land and $800,000/km under the river. To minimize the cost of the pipeline, to what point P on the south bank should the pipe come out of the river? (Draw a picture of the problem and show your point P. Note: time has nothing to do with this problem.)
Explanation / Answer
An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 3 km east of the refinery. The cost of laying pipe is $400,000/km over land to a point P on the north bank and $800,000/km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located?
P = 3-2/sqrt(3), roughly 1.85 km east of the refinery
Explanation:
The function defining the costs for the project is:
f(p)=400000*p+800000*sqrt(2^2+(3-p)^2)
where 400000*p is the costs for laying alongside the river to point p
and 800000*sqrt(2^2+(3-p)^2) is the costs for diagonally laying the pipe under the river from point P to the tanks (according to pythagoras)
(Try wolframalpha.com and type "plot f(p)=400000*p+800000*sqrt(4+(3-p)^2) from -10 to 10")
This function has a global minimum at p = 3-2/sqrt(3)