Answer
The water level is rising at a rate of $0.8~ft/min$
Work Step by Step
We can use similar triangles to write an expression for the distance $b$ across the top of the water in terms of the water level $h$:
$\frac{b}{h} = \frac{3}{1}$
$b = 3h$
We can differentiate both sides of the equation for the water volume with respect to $t$:
$V = \frac{1}{2} b~h~L$
$V = \frac{1}{2}(3h^2)(10)$
$V = 15~h^2$
$\frac{dV}{dt} = (30~h)~\frac{dh}{dt}$
$\frac{dh}{dt} = (\frac{1}{30h})(\frac{dV}{dt})$
$\frac{dh}{dt} = [\frac{1}{(30)(0.5)}](12)$
$\frac{dh}{dt} = 0.8~ft/min$
The water level is rising at a rate of $0.8~ft/min$
