Find

$$\int \frac{(4-3x{)}^2}{5\sqrt{x}}dx,x>0$$

giving your answer in simplest form.

The rate of change of the depth of water, $D$ metres, in a reservoir with respect to time $t$ days is modeled by the equation $\frac{dD}{dt}=t^2+at+b$ for $0\le t\le 10$, where $a$ and $b$ are constants.

• At the start of a monitoring period ($t=0$), the depth is 15 metres.
• After 3 days, the depth has fallen to 9 metres.
• On the 3rd day ($t=3$), the depth is decreasing at a rate of 2 metres per day.

Determine the expression for $D$ in terms of $t$, giving your answer in simplest form.