Speed of the Tip of a Shadow: A Mathematical Analysis

In this article, we delve into a classic problem involving the motion of the tip of a shadow cast by a man walking away from a street light. Using principles of similar triangles and geometry, we'll derive the mathematical relationship that determines the speed of the shadow's tip. This analysis is not only a fun exercise in applied mathematics but also a practical demonstration of the importance of geometric principles in real-world scenarios.

Setup of the Problem

A man, 6 feet tall, walks away from a street light at a constant speed of 5 feet per second. The street light is positioned on a 24-foot pole. We are tasked with finding how fast the tip of his shadow is moving along the ground when he is 50 feet from the pole. To solve this, we will employ the principles of similar triangles and analyze the relationship between the distances involved in the problem.

Step-by-Step Solution

Step 1: Define Variables

Let's denote the variables as follows:

h_m 6 feet, height of the man h_p 24 feet, height of the pole d 50 feet, distance of the man from the pole s length of the shadow of the man x distance from the pole to the tip of the shadow

Step 2: Set Up the Similar Triangles

The triangles formed by the pole and the shadow are similar to the triangle formed by the man and his shadow. Therefore, we can set up the following proportion:

(frac{h_p}{x} frac{h_m}{x-s})

Step 3: Cross-Multiply and Solve for (s)

Substituting the known heights, we have:

(frac{24}{x} frac{6}{x-s})

After cross-multiplying, we get:

24(x-s) 6x

Expanding and isolating (s), we have:

24x - 24s 6x

18s 18x

s (frac{18x}{18} x)

However, we need to express (s) in terms of (d).

s (frac{d}{3})

Step 4: Differentiate with Respect to Time

Now, we differentiate (s) with respect to time (t):

(frac{ds}{dt} frac{1}{3}frac{dd}{dt})

Given that the man is walking away at a speed of 5 feet per second, we have:

(frac{dd}{dt} 5) feet per second

Therefore:

(frac{ds}{dt} frac{1}{3} times 5 frac{5}{3}) feet per second

Step 5: Find the Velocity of the Tip of the Shadow

The speed of the tip of the shadow is given by the sum of the man's speed and the shadow's speed:

(frac{dx}{dt} frac{dd}{dt} frac{ds}{dt})

Substituting the values we have:

(frac{dx}{dt} 5 frac{5}{3} frac{15}{3} frac{5}{3} frac{20}{3}) feet per second

Therefore, the speed of the tip of the shadow is:

(frac{20}{3}) feet per second, approximately 6.67 feet per second.

Conclusion

This analysis demonstrates the mathematical beauty in solving real-world problems using principles of geometry and algebra. The speed of the tip of the shadow, despite the increasing distance, remains constant as long as the man walks at a constant speed. This problem not only illustrates the importance of similar triangles in practical scenarios but also provides a deeper understanding of how geometry can help us analyze and predict physical phenomena.