Answer
The length of the shadow is decreasing at a rate of $0.6~m/s$
Work Step by Step
Let $x$ be the man's distance from the spotlight.
Let $z$ be the height of the shadow.
We can use similar triangles to write an expression for $z$:
$\frac{z}{12} = \frac{2}{x}$
$z = \frac{24}{x}$
We can differentiate both sides of the equation with respect to $t$:
$z = \frac{24}{x}$
$\frac{dz}{dt} = \frac{-24}{x^2}~\frac{dx}{dt}$
$\frac{dz}{dt} = [\frac{-24}{(8)^2}~]~(1.6)$
$\frac{dz}{dt} = -0.6$
The length of the shadow is decreasing at a rate of $0.6~m/s$.
