Math optimization

**rmn** · November 12th 2007, 12:36 PM

http://i207.photobucket.com/albums/b...EvErSe_/22.jpg

I have absolutely no idea how to do this or even start it

**galactus** · November 12th 2007, 02:28 PM

Let k be the amount of light admitted per unit area of glass.

The total amount of light admitted by the whole window is:

$L=k(\text{area of clear glass})+\frac{1}{2}k(\text{area of colored glass})=2krh+\frac{1}{4}{\pi}kr^{2}$

But the perimeter is $P=2h+2r+{\pi}r$

$h=\frac{1}{2}(P-2r-{\pi}r)$

So,

$L=kr(P-2r-{\pi}r)+\frac{1}{4}{\pi}kr^{2}=k\left[Pr-(2+{\pi}-\frac{\pi}{4})r^{2}\right]=k\left[Pr-\frac{8+3\pi}{4}r^{2}\right]$

for $0<r<\frac{P}{2+{\pi}}$

Now, differentiate wrt r, set to 0 and solve for r.

**rmn** · November 12th 2007, 05:43 PM

sry what does L mean and isnt the rectangle area equal lw not rh?

**jwebb19** · November 12th 2007, 05:47 PM

L stands for Light. I believe he used r because the length of the rectangle is 2*radius of the semicircle. h is just the height of the rectangle.

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