Calculating the Rate at Which a Worker's Shadow Shortens as He Moves Towards a Lamp Post

Have you ever wondered how quickly a person's shadow shortens as they move towards a light source?

Problem Statement

Let's consider a specific scenario: A worker, who is 1.5 meters tall, is moving at a speed of 1.05 meters per second towards a lamppost that is 3.05 meters high. Given this setup, we need to determine how fast his shadow is shortening.

Using Similar Triangles

To solve this problem, we can use the concept of similar triangles. Let y be the length of the shadow and x be the distance of the person from the lamp post. Based on the similar triangles formed, we have:

1.5/y 3.05/xy

From this, we can derive:

xy/y 3.05/1.5

x/y 1.55/1.5

y 1.5/1.55x

or

dy/dt -1.5/1.55dx/dt

Given that dx/dt is 1.05 meters per second (as the person is moving towards the lamppost), we can calculate:

-dy/dt 1.5/1.55 - 1.05

-dy/dt 1.5/1.55 - 1.05 -1.01612820513 meters per second

Thus, the shadow is shortening at a rate of approximately 1.016 meters per second.

Lamp Post and Shadow Length at Night

Now, let's explore a different scenario where it's night time and the only source of light is from the top of the lamppost. Let's denote the height of the post by h_p and the height of the worker by h_w. Suppose h_p 3.05 meters and h_w 1.5 meters. Let the length of the shadow be L and the distance of the worker from the lamp post be D.

Using similar triangles again, we can write:

LD/L h_p/h_w 3.05/1.5 2.033

2.033L L D

1.033L D

L 0.968D

So, dL/dt is equal to 0.968 times dD/dt:

dL/dt 0.968(1.05) 1.016 meters per second

This means that the length of the shadow is growing at a rate of 1.016 meters per second, contrary to the scenario where the shadow was shortening.

Conclusion

The rate at which a worker's shadow shortens depends on the relative positions and heights of the worker and the lamp post. At night, the shadow length can increase, while during the day, it can decrease. Understanding these principles can be useful in various applications, such as analyzing shadows in photographs or understanding the dynamics of light and shadow in urban environments.