Answer
The angle between the ladder and the ground is decreasing at a rate of $~~\frac{1}{8}~rad/s$
Work Step by Step
Let $x$ be the horizontal distance between the bottom of the ladder and the wall. Let $\theta$ be the angle the ladder makes above the horizontal.
We can relate $x$ and $\theta$ in an equation. Then we can differentiate both sides with respect to $t$:
$\frac{x}{10} = cos~\theta$
$x = 10~cos~\theta$
$\frac{dx}{dt} = -10~sin~\theta~\frac{d\theta}{dt}$
$\frac{d\theta}{dt} = -\frac{csc~\theta}{10}\frac{dx}{dt}$
$\frac{d\theta}{dt} = -\frac{\frac{10~ft}{8~ft}~}{10}(1~ft/s)$
$\frac{d\theta}{dt} = -\frac{1}{8}~rad/s$
The angle between the ladder and the ground is decreasing at a rate of $~~\frac{1}{8}~rad/s$
