A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast is the man's shadow lengthening and how fast the tip of shadow is moving when he is walking away from the light at the rate of 100 ft/min.

Explanation

Let S be the position of source of light.
Let BD be the position of the man at a time t.
Let `A=x` and `BC=y=` length of the shadow.
Now, `(dx)/(dt)=100`ft./min.
From figure, `DeltaASC~DeltaBDC`

`:.(AS)/(BD)=(AC)/(BC)`
`(30)/(6)=(x+y)/(y)`
`:.5y=x+y`
`y=x/4`
`:.(dy)/(dx)=1/4*(dx)/(dy)=1/4xx100=25` ft./min.
`:. ` Shadow of the man is lengthening at the rate 25 ft./min.
The tip of the shadow is at C. Let AC = z.
`:.AB=x,BC=z-x`.
`(AS)/(BD)=(AC)/(BC)`
`:.(30)/(6)=(z)/(z-x)`
`:.5z-5x=z`
`:.z=(5)/(4)x`.
`:.(dz)/(dt)=5/4.(dx)/(dt)=(5)/(4)(100)`
`=125` ft./min.
`:.` Tip of the shadow is moving at the rate of 125 ft./min.