Calculus, Optimization: A fence is to be built to enclose an

[chucknorrisfish]

A fence is to be built to enclose a rectangular area of 260 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 16 dollars per foot. Find the length L and width W(with W <= L) of the enclosure that is most economical to construct.

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not quite sure how to set this up/ do this problem.

 

[tkhunny]

Name Stuff.
Find the various lengths of the fence.
Find the various costs of the lengths of the fence.
I just might lead to a quadratic with a negative leading coefficient.
Then what?

 

[chucknorrisfish]

OK...So i have A=LW...

260 = L W
L= 16 per foot
(W = 5 per foot) x 3

so 260 = 16(L) + 15w ? is that right?

 

[soroban]

Re: Calculus, Optimization: A fence is to be built to enclos

Hello, chucknorrisfish!

You aren't reading the problem carefully . . .

A fence is to be built to enclose a rectangular area of 260 square feet.
The fence along three sides is to be made of material that costs 5 dollars per foot,
and the material for the fourth side costs 16 dollars per foot.
Find the length \(\displaystyle L\) and width \(\displaystyle W\;(W\,\leq\,L)\) that is most economical to construct.

              L
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    W |               | W
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      * - - - - - - - *
              L

We are told that the area must be 260 ft². \(\displaystyle \;LW\;=\;260\;\;\Rightarrow\;\;W\:=\:\frac{260}{L}\;\) [1]

The front \(\displaystyle (L)\) and the two sides \(\displaystyle (2W)\) cost $5 per foot.
. . Their cost is: \(\displaystyle \,5(L\,+\,2W)\) dollars.

The back \(\displaystyle (L)\) costs $16 per foot.
. . Its cost is: \(\displaystyle \,16L\) dollars.

Hence, the total cost is: \(\displaystyle \,C\:=\:5(L\,+\,2W)\,+\,16L \:=\:21L\,+\,10W\;\) [2]

Substitute [1]: \(\displaystyle \:C\;=\;21L\,+\,10\left(\frac{260}{L}\right)\;\;\Rightarrow\;\;\fbox{C\;= \;21L\,+\,2600L^{-1}}\)

And that is the function we must minimize . . .