Difficult Extrema Word Problem

[UCdavisEcon]

I'm an econ major finishing up my sophomore year . I'm in calc 1. I am having trouble starting the problem

Problem: The intensity of light, such as the sun’s rays, depends on the angle at which the light meets the surface of the ground. The amount of illumination is proportional to the intensity of the light source. It is also jointly proportional to the reciprocal of the square of the distance from the light source and the sine of \(\displaystyle \, \theta\,\), where \(\displaystyle \, \theta\,\) is the angle at which the light strikes the ground.

A rectangular room with dimensions 10 ft by 24 ft has a 10-ft ceiling and a light is to be placed hanging from the center of the ceiling. Determine the height at which the light should be placed in order for the corners to receive as much light as possible.

 

[Deleted member 4993]

I'm an econ major finishing up my sophomore year . I'm in calc 1. I am having trouble starting the problem

Problem:

The intensity of light, such as the sun’s rays, depends on the angle at which the light

meets the surface of the ground. The amount of illumination is proportional to the
intensity of the light source. It is also jointly proportional to the reciprocal of the square
of the distance from the light source and the sine of  , where  is the angle at which
the light strikes the ground.
A rectangular room with dimensions 10 ft by 24 ft has a 10-ft ceiling and a light is to be
placed hanging from the center of the ceiling. Determine the height at which the light
should be placed in order for the corners to receive as much light as possible.

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

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[stapel]

Problem: The intensity of light, such as the sun’s rays, depends on the angle at which the light meets the surface of the ground. The amount of illumination is proportional to the intensity of the light source. It is also jointly proportional to the reciprocal of the square of the distance from the light source and the sine of \(\displaystyle \, \theta\,\), where \(\displaystyle \, \theta\,\) is the angle at which the light strikes the ground.

For this part, use what you learned back in algebra about setting up "variation" equations. Create a variation equation for the illumination (or "light") "L" in terms of the other variables.

For the sine part, draw a picture with, say, "d" as the "distance" (through the air) from the light source to the corner, "h" as the "height" above the floor of the light, and "x" as the distance on the floor from the point directly under the light to the corner. Label the angle at the corner as \(\displaystyle \, \theta\).

A rectangular room with dimensions 10 ft by 24 ft has a 10-ft ceiling and a light is to be placed hanging from the center of the ceiling.

What does the Pythagorean Theorem tell you then about the value of "x"?

Determine the height at which the light should be placed in order for the corners to receive as much light as possible.

You want to maximize L. You need to find the best (that is, the maximizing) value of h (the height). So try expressing the sine ratio in terms of x (whose value you now know) and h, using the definition of the sine ratio. See where this leads.

 

[UCdavisEcon]

Alright, thanks for the help.

So far I've come up with y= (kix)/(169-x^2)^(3/2)
where y Is the amount of illuimination, k is the constant of proportionality. x is the height directly beneath the light to the ground.

the attached picture Is a diagram of the problem. does it appear I have set it up correctly?

 

[Deleted member 4993]

Alright, thanks for the help.

So far I've come up with y= (kix)/(169-x^2)^(3/2)
where y Is the amount of illuimination, k is the constant of proportionality. x is the height directly beneath the light to the ground.

the attached picture Is a diagram of the problem. does it appear I have set it up correctly?

You have:

y= (kix)/(169 + x^2)^(3/2) ..... what is that i ? In the numerator you should have + sign.

Now differentiate y to get y' = f(x)

Set y' = 0 for maxima/minima

However, you'll get the answer quicker (and understand it better) - if you plotted y vs. x with your graphing calculator (or computer).

 

[Minivan]

What did you end up getting for your answer?

 

[UCdavisEcon]

"I" represents intensity.

 

[pepperdinepanda]

I am working on this problem, but I am not sure I differentiated correctly. What did you get when you differentiated to find y'?

 

[stapel]

I am working on this problem, but I am not sure I differentiated correctly.

Please reply showing your work and results. Thank you!

 

[Deleted member 4993]

I am working on this problem, but I am not sure I differentiated correctly. What did you get when you differentiated to find y'?

Please tell us what you did get for y', so that we can advise you if you did make any mistake.

 

[pepperdinepanda]

Please tell us what you did get for y', so that we can advise you if you did make any mistake.

Okay, so I differentiated using the quotient rule. I treated k and i as constants, but I came up with a big old mess of a fraction.
I'm sure I made a mistake somewhere.

y'=ki(4826809-85683x^2- 2x^5)/(169+x^2)^3*((169+x^2)^2/3)