A street light is at the top of a 13 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 7 ft/s along a straight path. How fast is the tip of her shadow moving when she is 40 feet from the base of the pole?

The tip of the shadow is moving at

(Include units in your answer)

13.00 ft/s

Step-by-step explanation:

Given

H = the height of the pole = 13 ft

h = the height of the woman = 6 ft

x = the length of the woman's shadow.

s = the distance from the pole to the woman = 40 ft

ds/dt = the constant rate at which the woman is walking = 7 ft/s

By similar triangles, H / h = (s + x) / x ⇒ H*x = h * (s + x).

Differentiating,

H*dx/dt = h * (ds/dt + dx/dt)

⇒ 13*dx/dt = 6 * (7 + dx/dt) = 42 + 6*dx/dt

⇒ 7 dx/dt = 42

dx/dt = 6 ft/s for how fast her shadow is lengthening.

The speed of the tip of her shadow would be this speed added to her traveling speed: 6 ft/s + 7 ft/s = 13.00 ft/s.