CALCULUS RELATED RATE PROBLEM. PLEASE HELP??

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 5ft/s along a straight path. How fast is the tip of the shadow moving when he is 40 ft from the pole?

1 Answer
sente
Sep 25, 2016

The tip of the shadow is moving at a speed of #25/3 = 8.bar(3)"ft"/"s"#

Explanation:

First, let's sketch the situation:

enter image source here

In the above image, #m# is the distance from the pole to the man, and #s# is the distance from the pole to the tip of the man's shadow. Our goal is to find the rate of change of #s# with respect to time given that rate of change of #m# with respect to time is #5"ft"/"s"# and #m=40"ft"#

As derivatives are rates of change, we can rewrite our goal as trying to find #(ds)/dt# given #(dm)/dt = 5# and #m=40#.

By the properties of similar triangles, we have

#s/15 = (s-m)/6#

#=> 2s = 5s - 5m#

#=>s = 5/3m#

Differentiating with respect to time, we get

#(ds)/dt = 5/3 (dm)/dt = 5/3*5 = 25/3#

As it so happens, the rate of change of the tip of the shadow with respect to time is independent of the value of #m#, and our final result is that the tip of the shadow is moving at a speed of #25/3 = 8.bar(3)"ft"/"s"#

Related questions
See all questions in Using Implicit Differentiation to Solve Related Rates Problems
Impact of this question
4345 views around the world
You can reuse this answer
Creative Commons License