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Calculus/Optimization

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Expert: - 3/12/2009


Question
A rancher wants to fence in an area of 3000000 square feet in a rectangular
field and then divide it in half with a fence down the middle parallel to one
side. What is the shortest length of fence that the rancher can use?

Thank you very much!
- James

Answer
The number 3000000 needs a couple of commas I think - I have a hard time counting 0's.  3,000,000 - Aha!
It's 3 million.

Now for the fence, look at the pasture on a piece of paper.
There is a top side side and a bottom side (that's two sides).
There are 3 sides the run vertically - one on either side and one in the middle.

Let's call the top and the bottom x.
Let's call the vertical ones (on the map, anyway) y.

The total distance is 2x + 3y.

We know that the area, 3,000,000, is defined by xy.
This says that y = 3M/x (you know M, a mega - that's a million).

The equation for length is then 2x + 9M/x.

There derivative of this function is 2 - 9M/x², which we set to 0 and solve for x.

That gives us x² = 9M/2 = 4,500,000.
The answer for x would be √(4.5) thousand.
The answer for y would be 3M/x.

The overall answer would be 2x + 3y.

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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