Answer
The length of the shadow is increasing at a rate of $400~ft/h$
Work Step by Step
Let $x$ be the length of the shadow.
Let $\theta$ be the angle of elevation of the sun.
$\frac{400}{x} = tan~\theta$
When $\theta = \frac{\pi}{6}$:
$\frac{400}{x} = tan~\theta$
$x = \frac{400}{tan~\theta}$
$x = \frac{400}{tan~\frac{\pi}{6}}$
$x = 693$
We can find $\frac{dx}{dt}$:
$\frac{400}{x} = tan~\theta$
$-\frac{400}{x^2}~\frac{dx}{dt} = sec^2~\theta~\frac{d\theta}{dt}$
$\frac{dx}{dt} = -\frac{x^2~sec^2~\theta~\frac{d\theta}{dt}}{400}$
$\frac{dx}{dt} = -\frac{(693)^2~(sec^2~\frac{\pi}{6})~(-0.25~rad/h)}{400}$
$\frac{dx}{dt} = 400~ft/h$
The length of the shadow is increasing at a rate of $400~ft/h$
