A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?

Question

Christopher Agresta · Accepted Answer

#20/3 (ft)/s#

Emilia Aguilar · Answer

undefined To find the rate at which the tip of the man's shadow is moving, we can use related rates. Let $ s $ be the distance between the man and the tip of his shadow, and $ x $ be the distance from the base of the light pole to the man.

$ \frac{{ds}}{{dt}} $ = rate of change of distance between man and tip of shadow
$ \frac{{dx}}{{dt}} $ = rate of change of distance from base of pole to man

Given:
$ \frac{{dx}}{{dt}} = 4 $ ft/s (rate at which the man is walking away from the pole)
$ x = 50 $ ft (distance from the base of the pole to the man)
Height of pole = 15 ft
Height of man = 6 ft

By similar triangles, we have:
$ \frac{{s}}{{x}} = \frac{{15}}{{6}} $
$ s = \frac{{5x}}{{2}} $

Differentiate both sides with respect to $ t $:
$ \frac{{ds}}{{dt}} = \frac{{5}}{{2}} \frac{{dx}}{{dt}} $

Substitute the given values and solve for $ \frac{{ds}}{{dt}} $:
$ \frac{{ds}}{{dt}} = \frac{{5}}{{2}} \times 4 = 10 $ ft/s

Therefore, the tip of the man's shadow is moving at a rate of 10 ft/s when he is 50 ft from the pole.