To solve this problem, we can use similar triangles. Let's denote the distance from the base of the pole to the woman as x, with x increasing as the woman walks away from the pole.
The height of the woman's shadow can be denoted as h, and the height of the pole can be denoted as H.
We are given:
- The height of the pole H is 15 feet.
- The height of the woman h is 6 feet.
- The rate at which the woman is moving away from the pole, dx/dt, is 7 ft/sec.
We need to find the rate at which the tip of the shadow, y, is moving away from the woman, dy/dt when x = 50 ft.
Now, let's set up the similar triangles:
First Triangle:
Side 1: woman's height h
Side 2: height of the pole H
Side 3: distance from the woman to the pole x
Second Triangle:
Side 1: height of the woman's shadow h
Side 2: height of the pole's shadow H + y (where y is the length of the shadow cast by the top of the pole)
Side 3: distance from the woman to the tip of the shadow x + y
Since the triangles are similar, we can set up the ratio:
h / H = (h + y) / (H + y)
Cross-multiplying, we get:
h(H + y) = H(h + y)
Expanding, we have:
hH + hy = hH + Hy
Simplifying, we obtain:
hy - Hy = hH - hH
This reduces to:
hy - Hy = 0
Now, let's differentiate both sides of the equation with respect to time, t:
d(yh)/dt - d(Hy)/dt = 0
Using the product rule and chain rule, we have:
y * dh/dt + h * dy/dt - H * dy/dt = 0
Since we know dh/dt (the rate at which the woman's height changes) is 0 (the woman's height remains constant), we can remove the y * dh/dt term:
h * dy/dt - H * dy/dt = 0
Factoring out dy/dt, we get:
( h - H ) * dy/dt = 0
Since dy/dt cannot be zero (it represents the rate at which the tip of the shadow is moving), we have:
h - H = 0
Substituting the given values:
6 - 15 = -9
Therefore, when the woman is 50 feet from the base of the pole, the tip of her shadow is not moving away from her. Thus, the tip of the shadow is moving at 0 ft/sec.