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Calculus/Maximum-minimum problem

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Question
A rectangular fence is to be put up using 2 kinds of fencing. 2 opposite sides will use heavy duty fencing and the other 2 will use standard fencing. Heavy duty costs $3 a foot and the standard costs $2 a foot. What are the dimensions of a rectangle with greatest area that will cost $6000?

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Questioner:Evan
Country:California, United States
Category:Calculus
Private:No
Subject:calculus

Question:

A rectangular fence is to be put up using 2 kinds of fencing. 2 opposite sides will use heavy duty fencing and the other 2 will use standard fencing. Heavy duty costs $3 a foot and the standard costs $2 a foot. What are the dimensions of a rectangle with greatest area that will cost $6000?
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Please check out:

http://en.allexperts.com/q/Calculus-2063/2009/11/Maximum-minimum-problem-41.htm

for lots of these examples.
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Start with:

Let x = the length of the side using HD fence.
Let y = the length of the side using ordinary fence.

Then

2x = HD fencing needed.
2y = std fencing needed.

and

6x = cost of HD fence
4y = cost of other.

Finally,

A = xy,  to maximize, subject to your constraint:

C = 6x + 4y = 6000.

The rest is routine:

Solve the constraint for x (or y).
Substitute into A.
Diff, set = 0, solve, etc.

You can finish up.

Calculus

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Paul Klarreich

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All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

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I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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