To understand how the shadow’s length changes as the man walks away from the light, we can use similar triangles. The man is 6 feet tall, and the light is 15 feet above the ground. As the man moves away from the light source, a triangle is formed between the top of the light, the tip of the shadow, and the ground. Simultaneously, a smaller triangle is created between the man and the tip of his shadow.
Let’s denote:
- h = height of the light = 15 feet
- m = height of the man = 6 feet
- s = length of the shadow
- d = distance of the man from the base of the light
By the properties of similar triangles, the ratios are equal:
h / (d + s) = m / s
Substituting the known heights:
15 / (d + s) = 6 / s
15s = 6(d + s)
Expanding and rearranging, we get:
15s = 6d + 6s
Taking like terms to one side:
15s - 6s = 6d
9s = 6d
So, we can express the shadow’s length:
s = (2/3)d
This means for every foot the man walks away from the light, his shadow grows by 2/3 of that distance. If he walks at a speed of 5 feet per second, his shadow will increase at a rate of:
(2/3) * 5 = 3.33 feet per second.
In summary, as the man moves away from the light, his shadow lengthens, allowing us to see how the relationship between the height of the light, the height of the man, and the distance traveled influences the shadow’s size.