# Illumination Problem

1. ### MC Escher

13
1. The problem statement, all variables and given/known data
I am going to use "Z" to represent theta,
The amount of illumination on a surface is given by I=kSinZ/d^2
where Z is the angle at which the light strikes the surface, k is the intensity of illumination (and is constant), and d is the distance from the light to a surface. A rectangular room measures 10 feet by 24 feet, with a 10-foot ceiling. Determine the height at which the light should be placed (in the center of the room) to allow the corners of the floor to receive the maximum amount of light.

2. Relevant equations
Optimization
Use of Derivatives
Differentials
All of these could be relevant, there aren't really any "equations" per se.

3. The attempt at a solution
I am quite sure you need to rewrite in terms of Z. I have a triangle set up with d as the hypotenuse, x the opposite side, and Z as the adjacent angle. I have:
SinZ=x/d
TanZ=x/13
d=13TanZ/SinZ
Here are some "hints" that were given:
1. Redefine I in terms of Z
2. Differentiate wrt Z to get 'the change in illumination wrt to the angle'
3. Determine Z when the differentiated expression equals O, and keep in mind the angle must be less than 90 degrees.
4. Determine x at this Z value
5. Differentiate a second time to verify if the value is a max or a min.

13
anyone? lol

3. ### Dick

25,910
Ok. What is the definition of 'maximum amount of light in a corner'? Do you want to maximize the sum of the intensities on each of the three surfaces meeting there? If so then you need to derive a formula relating the vertical height of the bulb to the three angles to each of these surfaces. Once you have done that then sum the three intensities and maximize wrt to the height. The lack of response here may be due to the fact that you didn't state a very definite 'question'. What part of this is presenting difficulties?

Last edited: May 17, 2007
4. ### MC Escher

13
I appreciate the help, but I solved the problem, 9.101