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"#

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