# Optimization Problem

1. Mar 19, 2006

### suspenc3

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 the 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?

Im having trouble setting up the problem..any help would be appreciated.

2. Mar 19, 2006

### Hootenanny

Staff Emeritus
What are you intial thoughts?

3. Mar 19, 2006

### suspenc3

I cant picture, or draw a diagram to illustrate the problem. I dont understand what the setup would look like.

4. Mar 19, 2006

### Hootenanny

Staff Emeritus
$$I = \frac{S}{r^2} + \frac{3S}{(10-r)^2}$$

Perhaps something like that?

5. Mar 19, 2006

### suspenc3

Yes, that makes sence, but what can I use to eliminate one of the variables?

6. Mar 19, 2006

### Hootenanny

Staff Emeritus
Before we proceed, it is probably more appropriate if we say that;

$$I\;\; \alpha \;\; \frac{S}{r^2} + \frac{3S}{(10-r)^2}$$

As, S is constant, the only two variables are I and r. We want to know how I varies with respect to r, if I understant the problem correctly?

So how about finding $I'(r)\;\;dr$?

7. Mar 19, 2006

### suspenc3

ok well, I found I' but I think it is wrong. I get

$$I' = \frac{-10r^2}{r^4} - \frac{30(r^2-2r+100)}{(r^2-2^r+100)^2}$$

When I try to solve for r I dont get a real number

8. Mar 19, 2006

### Hootenanny

Staff Emeritus
I get

$$I'(r) = \frac{6S}{(10-r)^3} - \frac{2S}{r^3}$$

You don't want to solve for I. What do you know about the gradient at a minimum point?

Last edited: Mar 19, 2006
9. Mar 20, 2006

### Hootenanny

Staff Emeritus
If you are having trouble visulaising the functions I have attached plot of them. The blue curve is that of the original function, the red is that of the derrivative. As S is constant(strength of source) I assigned an arbitray value of S=1 for this plot.

Hope this helps

#### Attached Files:

• ###### illumination.jpg
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10. Mar 14, 2011

### BeccaWoodard

Hey Suspenc, did you ever figure out this problem. I am having trouble figuring it out myself and was wondering if you could assist.