Question

Andy is 6 feet tall and is walking away from a street light that is 30 feet above ground at a rate of 2 feet per second. How fast is his shadow increasing in length?

Answer 1

`2.5` feet/sec

Explanation:

Let the man be `x` feet away from the street light, and his shadow length be `y`, Let the tip of the shadow be `l` feet away from the light such that `l=x+y` and we are given that `dx/dt=2`

By similar triangles:

`y/6=(x+y)/30`
`:. 30y=6(x+y)`
`:. 30y=6x+6y`
`:. 30y-6y=6x`
`:. 24y=6x`
`:. y=1/4x`

Now, `l=x+y`

`=> l=x+1/4x`
`:. l=5/4x`

Differentiating wrt `x` gives;

`(dl)/dx=5/4`

And by the chain rule we have:

`(dl)/dt=(dl)/dx dx/dt`
`:. (dl)/dt=5/4 xx 2`
`:. (dl)/dt=2.5` feet/sec