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?

1 Answer
Timber Lin
Dec 22, 2017

#20/3 (ft)/s#

Explanation:

enter image source here
in this diagram, x is the distance from the man to the pole, and y is the distance from the tip of the man's shadow to the pole. i assume the man and pole are standing straight up, which means the 2 triangles are similar.
by similarity, #(y-x)/y=6/15#
#15(y-x)=6y#
#15y-15x=6y#
#9y=15x#
#y=5/3x#

differentiate both sides with respect to #t# or time.
#dy/dt=5/3dx/dt#

you know #dx/dt=4(ft)/s# because the man is walking that speed away from the pole. you want to find #dy/dt#, how fast the tip of the shadow is moving.

that means #dy/dt=5/3*4(ft)/s=20/3(ft)/s#

actually, the man's distance from the pole doesn't matter since only his speed affects how fast his shadow moves.

Related questions
See all questions in Using Implicit Differentiation to Solve Related Rates Problems
Impact of this question
237539 views around the world
You can reuse this answer
Creative Commons License