# light intensity minimization

• Mar 20th 2009, 09:55 AM
mike_302
light intensity minimization
http://i160.photobucket.com/albums/t...2/untitled.jpg

My only thought on this: I have to minimize the l(x) equation it gives... But I can't figure out how to do that :S We know the power and quotient rule... no chain rule
• Mar 20th 2009, 12:02 PM
skeeter
Quote:

Originally Posted by mike_302
http://i160.photobucket.com/albums/t...2/untitled.jpg

My only thought on this: I have to minimize the l(x) equation it gives... But I can't figure out how to do that :S We know the power and quotient rule... no chain rule

$\displaystyle I = ax^{-2} + b(s-x)^{-2}$

$\displaystyle \frac{dI}{dx} = -2ax^{-3} + 2b(s-x)^{-3}$

$\displaystyle \frac{dI}{dx} = -\frac{2a}{x^3} + \frac{2b}{(s-x)^3}$

set the derivative equal to 0 ...

$\displaystyle \frac{2a}{x^3} = \frac{2b}{(s-x)^3}$

$\displaystyle bx^3 = a(s-x)^3$

$\displaystyle \sqrt[3]{b} \cdot x = \sqrt[3]{a} \cdot (s-x)$

$\displaystyle x = \frac{\sqrt[3]{a} \cdot s}{\sqrt[3]{a} + \sqrt[3]{b}}$