- The volume \(V \mathrm {~cm} ^ { 3 }\) of a spherical balloon with radius \(r \mathrm {~cm}\) is given by the formula
$$V = \frac { 4 } { 3 } \pi r ^ { 3 }$$
- Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) giving your answer in simplest form.
At time \(t\) seconds, the volume of the balloon is increasing according to the differential equation
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 900 } { ( 2 t + 3 ) ^ { 2 } } \quad t \geqslant 0$$
Given that \(V = 0\) when \(t = 0\)
- solve this differential equation to show that
$$V = \frac { 300 t } { 2 t + 3 }$$
- Hence find the upper limit to the volume of the balloon.
- Find the radius of the balloon at \(t = 3\), giving your answer in cm to 3 significant figures.
- Find the rate of increase of the radius of the balloon at \(t = 3\), giving your answer to 2 significant figures. Show your working and state the units of your answer.