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pandapatrol26

Member

Registered: 2007-04-03

Posts: 3

calculus problem: need help

Hi I have a calculus problem, and it would be great if you could give me some hints on how to solve it

Supertankers off-load oil at a docking facility 4 miles offshore. The nearest refinery is 9 miles east of the shore point nearest the docking facility. A pipeline must be constructed connecting the docking facility with the refinery. The pipeline costs $300,000 per mile if constructed underwater and $200,000 per mile if overland.

a) Locate Point B to minimize the cost of the construction.

b) The cost of underwater construction is expected to increase, whereas the cost of overland construction is expected to stay constant. At what cost does it become optimal to construct the pipeline directly to Point A.

                 Docking Facility
                   |\
                   |  \
            ^     l    \
4 miles  |      l      \
            |      l        \                                                          shore
            V     l--------\----------------------------------------------------

                   A          B             i put this as (9-x)      Refinery

                   |--------------------------9 miles -------------|

                  (i put the distance from A to B as X)

                 

* I solved question a by taking the hypotenuse = squareroot(16+X^2) and adding it to (9-x). Then I multiplied the hypotenuse by 300000 (since it's under water) and the (9-X) by 200000 since its on shore. I took the derivative of the entire equation:

1. 300000(sqroot(16+X^2) + (9-x)200000
2. f'(x) = (300000x)/(sqroot(X^2+16)) - 200000

Then I proceeded to solve for x:
X= sqroot(12.8)
X= 3.5777

Now I just need help with the second problem. Thanks

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: calculus problem: need help

For the first part, did you verify that your value of x does minimize (and not maximize) the cost?

So it will indeed minmize the cost. Well, substitute your x into the equation and find out what the minimum cost is. Say, this is C.

Now, the underwater building cost is going to rise. Let W be the underwater cost. Then as W increases (while the overland cost remains constant), x is going to have to decrease in order to keep the total cost constant at C. Eventually it will shrink down to 0. At this point, 4W + 1800000 = C. Now just solve for W.

Last edited by JaneFairfax (2007-04-03 09:30:19)

pandapatrol

Guest

Re: calculus problem: need help

For the first part, did you verify that your value of x does minimize (and not maximize) the cost?

So it will indeed minmize the cost. Well, substitute your x into the equation and find out what the minimum cost is. Say, this is C.

Now, the underwater building cost is going to rise. Let W be the underwater cost. Then as W increases (while the overland cost remains constant), x is going to have to decrease in order to keep the total cost constant at C. Eventually it will shrink down to 0. At this point, 4W + 1800000 = C. Now just solve for W.

I solved for W, but it ends up being less than 300,000. something like 223607. Is there something wrong with that?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: calculus problem: need help

Why, yes. My method was totally wrong, sorry. In fact, what was I thinking? If the underwater cost increases while the overland cost stays the same, the total cost must increase as well; it can’t stay the same. Argh! My fault.

So, let’s start all over again … let W be underwater cost. Then

If the cost is minimum at x = 0, then f′(0) = 0. But from the above equation, f′(0) cannot be 0. Therefore, I’d say the answer to the second part is that it’s impossible ever to build the underwater pipeline directly to A. That’s my tentative solution.

Last edited by JaneFairfax (2007-04-03 23:31:17)