13. The volume of a spherical balloon of radius \(r \mathrm {~m}\) is \(V \mathrm {~m} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)
- Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\)
Given that the volume of the balloon increases with time \(t\) seconds according to the formula
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 20 } { V ( 0.05 t + 1 ) ^ { 3 } } , \quad t \geqslant 0$$
- find an expression in terms of \(r\) and \(t\) for \(\frac { \mathrm { d } r } { \mathrm {~d} t }\)
Given that \(V = 1\) when \(t = 0\)
- solve the differential equation
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 20 } { V ( 0.05 t + 1 ) ^ { 3 } }$$
giving your answer in the form \(V ^ { 2 } = \mathrm { f } ( t )\).
- Hence find the radius of the balloon at time \(t = 20\), giving your answer to 3 significant figures.
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