Question

A man 2 meters tall walks at a rate of 2 meters per second toward a streetlight that is 5 meters tall. How fast is the tip of the man's shadow moving when he is 3 meters from the streetlight?

Answer 1

The problem is a related rates calculus problem where you find the tip of the man's shadow's speed by setting up a proportion related to the streetlight and the man.

Step-by-step explanation:

The question relates to a rate-of-change problem, which is typically addressed using principles from related rates in calculus. Here's how to approach the problem:

Step-by-Step Solution:

1. Let's denote the distance of the man from the streetlight as x (which is 3 meters initially) and the length of the shadow as y.
2. Given that the streetlight is 5 meters tall and the man is 2 meters tall, we can set up a proportion because the triangles formed are similar: 2/y = 5/(x + y).
3. Differentiate both sides with respect to time t to obtain the rates of change: d(2/y)/dt = d(5/(x + y))/dt.
4. Substitute the known values, including the man's speed (-2 m/s, where negative indicates he is approaching the streetlight), x = 3, and solve for dy/dt, which is the speed of the shadow's tip.
5. The final answer for the speed of the shadow's tip when the man is 3 meters from the streetlight will be a positive value, as the shadow's length is increasing.

The average speed checking step suggests that a result around 2.5 m/s is reasonable, so we should expect a number slightly above this for the speed of the shadow's tip.