
Multivariate Max and Min
A roof gutter is to be made from a long strip of sheet metal, 24 cm wide, by bending up equal amounts at each side through equal angles. Find the angle and the dimensions that will make the carrying capacity of the gutter as large as possible.
I'm not even sure how to set this up...

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A picture is worth a thousand words. (See attached image.)
I'll let you verify that the area of the figure is $\displaystyle A(x,a)=(242x+x\sin a)x\cos a$
Now maximize A by finding $\displaystyle x$ and $\displaystyle a$ such that $\displaystyle \frac{\partial A}{\partial x}$ and $\displaystyle \frac{\partial A}{\partial a}$ are zero.
(To check your work, I got $\displaystyle x=4.8cm$ and $\displaystyle a=30^\circ$)