Calculas H/W Help

Question

A street light is at the top of a 15 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 35 ft from the base of the pole?

Answer 1

Sketch a right triangle with vertical leg = 15 ft. At some point put a vertical 6 ft length inside this triangle. Label horizontal distance from light pole to the woman as "x", and the remaining distance to the vertex as "s". Set up the proportion between the two legs of similar triangles as s/6 = (x + s)/15, then 15s = 6x + 6s and 9s = 6x. Now differentiate: 9 ds/dt = 6 dx/dt, where dx/dt is given as 7 ft/s. So ds/dt = 42/9 = 14/3, and

the speed of the tip of the shadow is d/dt (x + s) = dx/dt + ds/dt = 7 + 14/3 = 35/3 ≅ 11.7 ft/sec.