# Maximum Area of a Rectangular Trough

I didn't learn how to do this in my class, and the examples in my book do not apply to this type of problem.

To make a rectangular trough, you bend the sides of a 32-inch wide sheet of metal to obtain the cross section pictured below. Find the dimensions of the cross section with the maximum area. (this will result in the trough with the largest possible volume).

I don't really know how to proceed. I know that I need to use derivatives to solve this, but I can't seem to get started

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Hint: let $h$ be the height of each side and $w$ be the width of the bottom. Can you write an equation based on the width of the strip? Can you write one for the area of the cross section? Solve the first for one variable, plug it into the second, and you will have an equation for the area that depends upon one variable. Take the derivative, set to zero....

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I just don't know how to write the equations when the only number that I have is the 32, I just can't figure it out. –  Hunter McMillen Nov 16 '11 at 17:49
Come on, if you know the height of each side surely you know how to write down an expression for the width. Append "$=w$" and voila! you have an equation. –  Henning Makholm Nov 16 '11 at 17:56
So $2h+w=32$, right? –  Ross Millikan Nov 16 '11 at 18:12
Yes I got that far but I am not understanding how I am supposed to get the length. Because I don't know what the measurement of the other side is. –  Hunter McMillen Nov 16 '11 at 18:39
The length doesn't matter. The question only asks about the cross sectional area, which we want to maximize. So what is the area? –  Ross Millikan Nov 16 '11 at 20:30

$$A = X\times W$$

$$32 = 2X + W$$

$$W = 32 - 2X$$

$$A = X(32 - 2X) = -2X^2 + 32X$$

$$\frac{dA}{dx} = -4X + 32$$

Set above equal to $0$ and solve

$$-4X + 32 = 0$$

$$X = \frac{32}{4} = 8$$

$$W = 32 - 16 = 16$$

$$A = W\times X = 16\times 8 = 128$$

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It would have been better if you had included more explanatory text in your (correct) answer. For example, "Let $A$ denote the cross-sectional area of the trough, $W$ be its width, $X$ be its height, then <your first equation>." Continue with "Now we know that <your second equation>" and so on, until the conclusion. –  Rick Decker Sep 22 '12 at 18:55