Let’s break down the scenario: a man who is 6 feet tall is walking away from a light located 15 feet high above the ground at a speed of 5 feet per second. To understand the relationship between the distance he walks and the light’s height, we can use some basic geometry and trigonometry.
1. **Understand the Setup**: The man is moving away from a light source at height. The distance of the man from the vertical line of the light source increases over time as he walks.
2. **Using Right Triangle Properties**: We can visualize the situation as forming a right triangle where:
- The height of the light is one side (15 feet).
- The distance from the man to the base of the light source is the other side (let’s call this d). This is what we want to find out.
- The line from the top of the light to the man forms the hypotenuse.
3. **Finding the Relationship**: As the man walks away, his distance (d) increases over time. If we consider time t (in seconds), the distance d from the light changes according to the formula:
d = 5t
4. **Calculating the Hypotenuse**: The distance from the man’s head to the light source can also be found using the Pythagorean theorem:
Length of hypotenuse = √(d² + (15 – 6)²)
This simplifies to:
Length of hypotenuse = √(d² + 9)
5. **Final Insights**: As the man continues to walk away at a pace of 5 feet per second, the distance d continues to increase linearly. At each moment in time, you can calculate the distance using the formulas provided above, keeping in mind that the height of the light above the ground is a constant 15 feet while the height of the man is 6 feet.
This relationship helps not only understand the spatial dynamics at play but also can serve practical applications, such as in safety assessments for lighting and visibility. A good visual representation can further enhance understanding of such situations.