To solve this problem, we can use similar triangles and the concept of proportions. Let's break it down step by step:
1. First, let's draw a diagram to visualize the situation. We have a woman who is 6 feet tall walking away from a light source that is 10 feet above the ground. As she moves, her shadow lengthens.
(Light Source)
-------------
| |
---------------- <- Ground
| Woman |
| |
----------------
2. Since the sun's rays are parallel, we have similar triangles formed by the woman, her shadow, and the ground. Let's denote the height of the woman as h_w, the length of her shadow as s, and the distance she walks as x. The height of the light source is h_l.
L____________
| |
| |
| _______ |
|| |
|| s |
||_________|
H x
3. By using similar triangles, we can establish the following proportion:
(h_w + h_l) / s = h_w / x
The numerator (h_w + h_l) represents the total height of the woman and the light source, while the denominator represents the distance the woman walks. Since the shadow lengthens at a rate of 2 ft/s, the derivative of s with respect to time (ds/dt) is equal to 2 ft/s.
4. Now, we can differentiate the proportion with respect to time (t):
d/dt [(h_w + h_l) / s] = d/dt (h_w / x)
5. We can simplify the left side of this equation by applying the quotient rule:
[s * (0 - (h_w + h_l)) / s^2] * ds/dt = (-h_w / x^2) * dx/dt
Simplifying further, we get:
- (h_w + h_l) * ds/dt / s^2 = -h_w * dx/dt / x^2
6. Since ds/dt = 2 ft/s, we can substitute it into our equation:
- (h_w + h_l) * 2 / s^2 = -h_w * dx/dt / x^2
7. Rearranging the equation to solve for dx/dt (the rate at which the woman is walking):
dx/dt = (s^2 * h_w * 2) / (x^2 * (h_w + h_l))
8. Now we can substitute the given values into the equation. The woman's height (h_w) is 6 feet, the light source's height (h_l) is 10 feet, and ds/dt is 2 ft/s:
dx/dt = (s^2 * 6 * 2) / (x^2 * (6 + 10))
9. We also know that s = x, because the triangles are similar. Therefore, we can simplify the equation further:
dx/dt = (x^2 * 6 * 2) / (x^2 * (6 + 10))
10. Canceling out the x^2 terms, we get:
dx/dt = 12 / 16
Simplifying, we have:
dx/dt = 3 / 4 ft/s
Therefore, the woman is walking at a rate of 0.75 ft/s.