A streetlight is 15 feet above the sidewalk. A man 6 feet tall walks away from the light at the rate of 5 ft/sec.

Question

a) Determine a function relating the length of the man’s shadow to his distance from the base of the streetlight. b) Determine the rate at which the man’s shadow is lengthening at the moment that he is 20 feet from the base of the light.

Answer 1

Create a diagram with a smaller right triangle embedded in a similar, bigger right triangle. The triangles share the acute angle created by the horizontal ground and the segment that connects the tip of the man’s shadow, the top of the man’s head, and the top of the streetlight. Label the man’s horizontal distance from the light x and the horizontal distance from his feet to the tip of his shadow, ie the length of his shadow, y.

By similar triangles,

y / 6 = (x+y) / 15

15y = 6x + 6y

9y = 6x

y = (2/3)x

dy/dt = (2/3)dx/dt. Since dx/dt = 5 , dy/dt = 10/3 ft/sec.

The rate at which the length of the shadow is increasing is independent of the man’s distance from the light.