A spherical drop of industrial lubricant is being injected into a precision-engineered cavity. The volume, $V$, of the drop is increasing at a constant rate of $180\pi {mm}^3{s}^{-1}$. Calculate the rate of increase of the radius, $r$, of the drop in ${mm\; s}^{-1}$ at the moment when the radius is exactly $3mm$. [The volume $V$ of a sphere of radius $r$ is given by $V=\frac{4}{3}\pi r^3$]

The depth of sediment, $ymetres$, settling at the bottom of an industrial filtration tank is monitored. The rate of change of the depth of the sediment is modeled by the differential equation

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

• at the start of monitoring ($t=0$), the sediment depth was $2metres$.
• after $5$ hours of monitoring, the sediment depth had reached $4metres$.
• after $T$ hours of monitoring, the sediment depth reached $6metres$.

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