HELP Related Rates Question: Light/Shadow

In summary, the conversation is about a calculus question involving a man walking towards a building with a light on the ground. The question asks for the rate at which the man's shadow on the building becomes shorter when he is 20ft away from the building. The solution involves using triangle ratios and differentiating to find the answer of -3.6ft/s. The conversation also includes a comment about giving hints instead of complete solutions.
  • #1
jen333
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0
Hey, I have this one practise calculus question that I just can't seem to get. Any help would be greatly appreciated:

A light is on the ground 40ft from a building. A man 6ft tall walks from the light towards the building at 6ft/s. How rapidly is his shadow on the building becoming shorter when he is 20ft from the building?
(btw, the answer should be -3.6ft/s)

I've already drawn a diagram, but I'm not sure if triangle ratios would do any good.

(oops, wrong forum AGAIN! I'd delete this is i could. Refer to calculus and beyond. Sorry)
 
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  • #2
jen333 said:
Hey, I have this one practise calculus question that I just can't seem to get. Any help would be greatly appreciated:

A light is on the ground 40ft from a building. A man 6ft tall walks from the light towards the building at 6ft/s. How rapidly is his shadow on the building becoming shorter when he is 20ft from the building?
(btw, the answer should be -3.6ft/s)

I've already drawn a diagram, but I'm not sure if triangle ratios would do any good.

(oops, wrong forum AGAIN! I'd delete this is i could. Refer to calculus and beyond. Sorry)
I'll transfer it for you.

I just wrote out a complete solution thinking the light was on the building and his shadow on the ground! Anyway the ratios are what you want but be careful about exactly what they are. Let x be the length of the man's shadow, on the building, and y be his distance from the light.
You have two similar right triangles:
1) The triangle formed by the man, line from the man to the light, and the hypotenuse. The ratio of second leg to first is y/6.
2) The triangle formed by the shadow of the man on the building, the line from the light to the base of the building, and its hypotenuse. The corresponding ratio is 40/x.
Since those triangles are similar, y/6= 40/x or xy= 240. Differentiating both sides, x'y+ xy'= 0. You are told that y'= 8 and you want to find x' when y= 20. Of course, then 20x= 240 so x= 12. 20x'+ 12(6)= 0. Solve for x'.
 
  • #3
Halls, I thought you weren't supposed to give complete solutions...I had written on this already giving her a hint... Also that is making it way too complicated. just write y = 240/x, and differentiate from there and plug in the values.
 
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  • #4
Thanx

Oh, hahaha. don't worry. I solved it before I saw the written out solution. Thanks so much for your help :D

-jen
 

1. What is a related rate question in regards to light and shadow?

Related rate questions in regards to light and shadow involve determining the rate at which the length, height, or angle of a shadow is changing in relation to the movement of an object or light source.

2. How do I approach solving a related rate question involving light and shadow?

First, identify the variables involved and their relationships. Then, use known formulas and trigonometric functions to set up an equation expressing the relationship between the variables. Finally, take the derivative with respect to time and plug in the given values to solve for the desired rate of change.

3. What are some common types of related rate questions involving light and shadow?

Some common types of related rate questions involving light and shadow include finding the rate of change of the length or height of a shadow as an object moves closer or farther away from a light source, or finding the rate of change of the angle of a shadow as the sun moves across the sky.

4. How does the position of the light source affect the rate of change in a related rate question involving light and shadow?

The position of the light source can significantly impact the rate of change in a related rate question involving light and shadow. For example, if the light source is directly overhead, the rate of change of the length of the shadow will be constant, whereas if the light source is at a lower angle, the rate of change will be more rapid.

5. Can related rate questions involving light and shadow be applied to real-world situations?

Yes, related rate questions involving light and shadow have many real-world applications, such as determining the optimal placement of solar panels for maximum efficiency or calculating the rate of change of shadows during a solar eclipse.

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