# Math Help - optimization

1. ## optimization

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 4 times as strong as the other, are placed 20 feet apart, where should an object be placed on the line between the sources so as to receive the least illumination?

2. Hello, singh1030!

I get an ugly cubic to solve . . .

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 4 times as strong as the other, are placed 20 feet apart,
where should an object be placed on the line between the sources
so as to receive the least illumination?

Let $S$ = strength of the light source
and $d$ = distance fronm the light source.

We are told that the illumination is: . $I \:=\:\frac{kS}{d^2}$
Code:
     (S)                         (4S)
* - - - - * - - - - - - - - *
A    x    P      20-x       B

Let $A$ be the source with strength $S$
and $B$ be the source with strength $4S.$

The object is placed at $P$, which is $x$ feet from $A$ and $(20-x)$ feet from $B.$

The object's illumination from $A$ is: . $I_A \:=\:\frac{kS}{x^2}$
The object's illumination from $B$ is: . $I_B \:=\:\frac{k(4S)}{(20-x)^2}$

The total illumination is: . $I \;=\;\frac{kS}{x^2} + \frac{4kS}{(20-x)^2} \;=\;kS\left[x^{-2} + 4(20-x)^{-2}\right]$

. . and that is the function we must minimize . . .