A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the bu

Question

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.4 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?

Answer 1

The shadow's height, as a proportion of the man's height, is equal to the shadow's distance from the light (12 m), divided by the man's distance from the light. Let x be the man's distance from the light. Then the height of his shadow is 12/x.

The rate of change in the shadow's height relative to a change in x is the derivative of 12/x :
d/dx (12/x) = d/dx (12x^-1) = -12x^-2 = -12/(x^2)

So when he is 4 m from the building, the rate of change in the shadow's height is __________ meters per meter traveled. (Just plug 4 into the above formula, and you can fill in the blank. Don't forget the minus sign.)

But that is a rate per meter traveled, not per second. So multiply it times the man's walking speed (1.4 m/s), and you'll have your answer in terms of meters (of change in the shadow's height) per second.

Answer 2

thank you everyone for answering!