The volume, $V$, of a spherical weather balloon is increasing at a constant rate of $120\pi {cm}^3{s}^{-1}$. Find the rate of increase of the radius, $r$, of the balloon in ${cm\; s}^{-1}$ at the instant when the radius is $6cm$. [The volume $V$ of a sphere of radius $r$ is given by the formula $V=\frac{4}{3}\pi r^3$]

The height of a pile of sand, $hmetres$, under a conveyor belt is monitored over time. The rate of increase in the height of the pile is modeled by the differential equation

where $k$ is a positive constant and $t$ hours is the time after the measurement began. Given that:

• initially (at $t=0$), the height of the pile was $3metres$.
• $4$ hours after monitoring began, the height of the pile was $5metres$.
• $T$ hours after monitoring began, the height of the pile was $8metres$.

Solve the differential equation to find the value of $T$. Give your answer to one decimal place.