14. The volume of a spherical balloon of radius \(r \mathrm {~cm}\) is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)
- Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\)
The volume of the balloon increases with time \(t\) seconds according to the formula
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 9000 \pi } { ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$
- Using the chain rule, or otherwise, show that
$$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$
where \(k\) and \(n\) are constants to be found.
Initially, the radius of the balloon is 3 cm .
- Using the values of \(k\) and \(n\) found in part (b), solve the differential equation
$$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$
to obtain a formula for \(r\) in terms of \(t\).
- Hence find the radius of the balloon when \(t = 175\), giving your answer to 3 significant figures.
(1) - Find the rate of increase of the radius of the balloon when \(t = 175\). Give your answer to 3 significant figures.