10 A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40 \pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8 \pi r\). The balloon remains a sphere at all times.
- Show that \(r\) and \(t\) satisfy the differential equation
$$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { 50 - r } { 5 r ^ { 2 } }$$
- Find the quotient and remainder when \(5 r ^ { 2 }\) is divided by \(50 - r\).
- Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\).
- Find the value of \(t\) when the radius of the balloon is 12 .