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 1.3 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? (Round your answer to one decimal place.)

Answer 1

decreasing 0.5 m/s

Explanation:

You want the rate at which the height of a shadow on a wall is decreasing as a 2 m tall man walks at 1.3 m/s toward the wall. The shadow is cast by a light 12 m from the wall, and the man is 4 m from the wall at the time of interest.

Similar triangles

In the attached diagram, triangles ABC and LMA are similar. This lets us write the proportion ...

y/x = 2/(12 -x)

y = 2x/(12 -x) . . . . . . multiply by x to get expression for y

Rate of change

Taking the derivative with respect to time gives ...

y' = (2/(12 -x) +2x/(12-x)²)x'

At x = 4, this is ...

y' = (2/8 +8/8²)x' = (3/8)x'

When x' = -1.3 m/s, the rate of change of the shadow height is ...

y' = (3/8)(-1.3 m/s) = -0.4875 m/s

The length of the shadow is decreasing at about 0.5 m/s.

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