(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Illumination Problem

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