# Calculus

A man of height 1.7 meters walks away from a 5-meter lamppost at a speed of
1.6 m/s. Find the rate at which his shadow is increasing in length. (Round your answer to three decimal places.) This problem is absolutely killing me I see a lot of examples like this on here but I can not seem to follow the steps correctly nor understand what is going on.

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1. 0.824 m/s

I figured it out hopefully this helps some of you. So draw your picture out and label the distance between the man and the pole as Y and the distance between the man and the shadow tip as X. Now use the properties of similar triangles to set up a proportion and then cross multiply.
5/(Y+X)=1.7/X
5X=1.7Y+1.7X
3.3X=1.7Y
Then use we need to find dx/dt! So start differentiating then isolate dx/dt.
3.3 (dx/dt)=1.7 (dy/dt)
(dx/dt)=(1.7(dy/dt))/3.3
Now just plug in the value they gave for the mans speed, this happens to be dy/dt.
The answer comes out to be 0.824 m/s.
Hope that helps!

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2. Okay. If the man is x meters from the pole, and his shadow's length is s meters, then using similar triangles, we have

(x+s)/5 = s/1.7
So, that means that
x = 5s/1.7 - s
or, s = 0.51x

So, ds/dt = 0.51 dx/dt = 0.51*1.6 = 0.82 m/s

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