Multivariate Max and Min

**Aryth** · October 20th 2009, 01:46 PM

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...

**Media_Man** · October 22nd 2009, 07:55 AM

A picture is worth a thousand words. (See attached image.)

I'll let you verify that the area of the figure is $A(x,a)=(24-2x+x\sin a)x\cos a$

Now maximize A by finding $x$ and $a$ such that $\frac{\partial A}{\partial x}$ and $\frac{\partial A}{\partial a}$ are zero.

(To check your work, I got $x=4.8cm$ and $a=30^\circ$ )

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