Calculus Question

Question

A street light is mounted at the top of a 15 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 4 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 50 ft from the base of the pole?

Answer 1

make a diagram and note that 15/DS=6/(DS-DW) where DS is distance of shadow tip from pole and DW is distance of shadow tip from woman. Rearrange so that DS=5/3DW and take derivative of both sides to find dDS/dt=5/3 dDW/dt. Note that we want to know dDs/dt and dDW/dt=4, thus dDS/dt=20/3 f/s.

Answer 2

If D = the distance from the pole and L = the length of the shadow, it can be shown that

L = (2/3)D, which means that

dL/dt = (2/3)dD/dt

Since dD/dt = 4 ft/sec at any distance from the pole

dL/dt = (2/3)(4) = 8/3 ft/sec at any distance from the pole

(From the pole, her shadow is moving 4 + 8/3 = 20/3 ft/sec)