How fast is the length of the shadow on the building decreasing?

Question

A spotlight on the ground shines on a wall 40 feet away. A 6 foot tall man walks from the spotlight toward the building at a rate of 5 ft/sec. How fast is the length of his shadow on the building decreasing when he is 10 feet from the building?

Answer 1

Labeling the horizontal axis [0,40] and given that the rate of change (or dx) is 5 fps, y = 6 and recall that x² + y² = c².

 

Take the derivative implicitly to get 2x dx + 2y dy = 0. At a distance of 10 feet from the wall, x = 30, y = 6, you get

 

2(30)(5) +2(6)(dy) = 0

 

300 + 12 dy = 0

 

12 dy = -300

 

dy = -12 fps

 

The length of the shadow on the building When the man is 10 feet from the building is decreasing at a rate of 12 fps.