# Thread: optimization? Need urgent help

1. ## optimization? Need urgent help

I am stumped with this question... Its due monday so any help i can get would be greatly appreciated.

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10ft apart, where should an object be placed on the line between the sources so as to receive the least illumination?

2. This one is a little different than the other illuminations problems I have seen.

Well, let's have a go.

Let S be the illumination at a point P which is, say, x units from the first source.

Let the two sources be $I_{1}, \;\ I_{2}$.

$S=\frac{kI_{1}}{x^{2}}+\frac{kI_{2}}{(10-x)^{2}}$

$\frac{dS}{dx}=\frac{-2kI_{1}}{x^{3}}+\frac{2kI_{2}}{(10-x)^{3}}=0$ when $\frac{2kI_{1}}{x^{3}}=
\frac{2kI_{2}}{(10-x)^{3}}$

$\frac{\sqrt[3]{I_{1}}}{\sqrt[3]{I_{2}}}=\frac{x}{10-x}$

$(10-x)\sqrt[3]{I_{1}}=x\sqrt[3]{I_{2}}$

Can you finish solving for x?. Remember that $I_{1}=3I_{2}$

or $3I_{1}=I_{2}$. Whatever. The problem does not specify.

That way you can eliminate a variable.

You can use the 2nd derivative test to check the point to see if it is the minimum, but it most likely will be.

3. thanks a lot you saved the day