Question

A street light is at the top of a 19 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 50 feet from the base of the pole?

Answer 1

The speed at which the tip of a shadow moves, related to a moving person and a light source, is found by solving a Related Rates problem. Using the principles of similar triangles, we can calculate the speed of the shadow's tip when the woman is 50 feet away to be 15.83 ft/sec.

Step-by-step explanation:

The student's question involves a Related Rates problem in Mathematics, where the goal is to determine how fast the tip of a shadow is moving when certain conditions are met. Let's denote the height of the pole as H, the height of the woman as h, the distance of the woman from the pole as x, and the length of her shadow as s. Using similar triangles, the relationship between the lengths and heights is H/h = (x+s)/x. Differentiating both sides with respect to time t gives us dH/dt (0, since the pole's height doesn't change) on one side and (1/h)(dx/dt)(s+x) + (x/(h^2))(ds/dt)(h) - (1/h)(dx/dt)(x) on the other. Simplifying, we find (ds/dt) = h/H(dx/dt) - (dx/dt). Using the given values, H = 19, h = 6, dx/dt = 5, and x = 50, we can calculate ds/dt.

Thus, the shadow length is growing faster than the woman walks since it includes the length of the woman's movement plus the 'extra' length of the shadow as the angles change. Solving this, we find the tip of the shadow moves away from the pole at the speed of 15.83 ft/sec when the woman is 50 feet from the pole.