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.

Transcript

in this lesson we're going to focus on the shadow problem in related rates so we have a six foot man and he walks at a rate of three feet per second away from a street light and his street light is 21 feet tall now let's draw a picture so let's say that's the ground and here is the street light now let's say this is the person now let's draw a line... Read More

Key Insights

• ☠️ Implicit differentiation is a useful technique for solving related rate problems.
• 😫 Similar triangles play a crucial role in setting up proportions to relate different lengths.
• ☠️ The rate of change of the shadow's length is mathematically related to the rate of change of the distance between the man and the light.
• ☠️ Differentiating implicitly allows us to find the rates of change for various components in the problem.

Questions & Answers

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.