To determine how fast the tip of the woman's shadow is moving, we can use similar triangles. Let's break down the problem and establish some variables.
Let:
- The height of the pole be represented by "h" = 17 ft.
- The distance between the woman and the pole be represented by "x" = 40 ft.
- The height of the woman be represented by "y" = 6 ft.
We can set up a proportion between the height of the pole, the height of the woman, and the height of their respective shadows:
h + y x
------- = ---
h s
Where "s" represents the height of the shadow.
By cross-multiplying, we can rewrite the equation:
(h + y)s = hx
Now, we'll differentiate both sides of the equation with respect to time (t) and apply the Chain Rule:
(d/dt)[(h + y)s] = (d/dt)[hx]
(h + y) * ds/dt = h * dx/dt
We are given that dx/dt = 6 ft/sec (since the woman is moving away at a speed of 6 ft/sec). Substituting all the known values into the equation:
(17 + 6) * ds/dt = 17 * 6
23 * ds/dt = 102
Finally, we can solve for ds/dt, the rate at which the tip of the shadow is moving:
ds/dt = 102 / 23 ≈ 4.43 ft/sec
Therefore, the tip of the woman's shadow is moving at a rate of approximately 4.43 ft/sec when she is 40 ft from the base of the pole.