- A large spherical balloon is deflating.
At time \(t\) seconds the balloon has radius \(r \mathrm {~cm}\) and volume \(V \mathrm {~cm} ^ { 3 }\)
The volume of the balloon is modelled as decreasing at a constant rate.
- Using this model, show that
$$\frac { \mathrm { d } r } { \mathrm {~d} t } = - \frac { k } { r ^ { 2 } }$$
where \(k\) is a positive constant.
Given that
- the initial radius of the balloon is 40 cm
- after 5 seconds the radius of the balloon is 20 cm
- the volume of the balloon continues to decrease at a constant rate until the balloon is empty
- solve the differential equation to find a complete equation linking \(r\) and \(t\).
- Find the limitation on the values of \(t\) for which the equation in part (b) is valid.