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:

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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.

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