Solve Illumination Problem: Use Derivatives & Differentials

  • Thread starter MC Escher
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In summary, the light should be placed at a height of 10 feet to allow the corners of the room to receive maximum light.
  • #1
MC Escher
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Homework Statement


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.


Homework Equations


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



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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  • #2
anyone? lol
 
  • #3
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:
  • #4
Dick said:
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?
I appreciate the help, but I solved the problem, 9.101
 

1. What is the illumination problem?

The illumination problem is a mathematical problem that involves finding the optimal way to distribute light in a given space. It is often encountered in engineering and design fields, where the goal is to achieve the most efficient and effective lighting solution.

2. How can derivatives be used to solve the illumination problem?

Derivatives can be used to solve the illumination problem by calculating the rate of change of light intensity at a specific point in the space. This allows us to optimize the placement and intensity of light sources to achieve the desired lighting conditions.

3. What is the role of differentials in solving the illumination problem?

Differentials are used to approximate the change in light intensity at a specific point in the space. By using differentials, we can break the problem into smaller, more manageable parts and apply the principles of calculus to find the optimal solution.

4. Are there any limitations to using derivatives and differentials to solve the illumination problem?

While derivatives and differentials can provide accurate solutions to the illumination problem, there may be limitations in certain scenarios. For example, if the space has complex geometry or if there are multiple light sources with varying intensities, the problem may become more challenging to solve using these methods.

5. How can the results of solving the illumination problem using derivatives and differentials be applied in real-world situations?

The results of solving the illumination problem can be applied in various real-world situations, such as designing efficient lighting systems for buildings, optimizing street lighting for safety and energy conservation, or improving the visibility of objects in photography and cinematography. These techniques can also be used in computer graphics to achieve realistic lighting effects in virtual environments.

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