Related Rates - The Shadow Problem | Summary and Q&A

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March 2, 2018
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The Organic Chemistry Tutor
Related Rates - The Shadow Problem

TL;DR

A man walking away from a street light raises questions about the changing length and tip of his shadow.

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Q: How can the rate at which the length of the shadow changes be calculated?

By setting up a proportion between two similar triangles formed by the man, the street light, and the shadow. The resulting equation can be differentiated implicitly to find the rate of change, which is 1.2 feet per second.

Q: How can the rate at which the tip of the shadow moves be determined?

By using similar triangles and setting up a proportion, the equation can be differentiated implicitly to find the rate of change. When the man is ten feet from the light, the tip of his shadow moves at a rate of 4.2 feet per second.

Q: What happens to the rate of change of the shadow's length if the man walks towards the light?

If the man walks towards the light, the rate of change of the shadow's length would be negative. The shadow would decrease in length at a rate of 1.2 feet per second.

Q: How does the height of the street light affect the rates of change?

The height of the street light does not directly affect the rates of change for the length or tip of the shadow. The rates of change are determined by the man's distance from the light.

Summary & Key Takeaways

• A man, six feet tall, walks at a rate of three feet per second away from a 21-foot-tall street light.

• Part A: As the man moves eight feet from the light, the length of his shadow changes at a rate of 1.2 feet per second.

• Part B: When the man is ten feet from the light, the tip of his shadow moves at a rate of 4.2 feet per second.