errr...calculus help?

oOLaura

A shelter for use at the beach has a back, two sides, and a top made of canvas. Find the dimensions that maximize the volume and require 96 square feet of canvas.

Please show work. Thanks in advance.

Answer
The key to this problem is to realize that the two "sides" of the shelter will be square. This is kind of hard to explain, I worked it out on paper and found that the variables representing the dimensions of the "sides" of the shelter could be swapped around without changing any of the formulae.

From here it is a simple maximization problem.
I'll do most of it for you:

Volume = A x B x C where A, B & C are the height, depth and width of the shelter respectively.
Area = 2 x A x B + A x C + B x C = 96

If A = B as I described above:
Area = 2A^2 + 2AC = 96
Therefore, C = (48 - A^2)/A
Therefore, Volume = A x ( 48 - A^2)

Now, maximize volume with respect to A.

Is this for high school? Regards,
Daniel

The canvas wind shelter is to be constructed for use on Padre Island?

Mixed Asia

beaches. It is to have a back, two square
sides, and a top. If 486 square feet of canvas
are available, find the length of the shelter for
which the space inside is maximized assuming
all the canvas is used.

Answer
call dimensions x,x, and z since square sides
486 = 2x^2 + 2xz ===> z=(243-x^2) / x ===>z=(243/x) - x
Maximize Volume by setting V'=0
V = x^2*z ===> plug in z
V = x^2(243/x - x)
V = 243x - x^3
V' = 3x^2 +243 = 0
x^2 = 81
x=9 (which is the square side)
z is the length z=(243/x) - x=243/9 - 9=27-9=18

18 feet

:)

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