- A balloon is being inflated.
In a simple model,
- the balloon is modelled as a sphere
- the rate of increase of the radius of the balloon is inversely proportional to the square root of the radius of the balloon
At time \(t\) seconds, the radius of the balloon is \(r \mathrm {~cm}\).
- Write down a differential equation to model this situation.
At the instant when \(t = 10\)
- the radius is 16 cm
- the radius is increasing at a rate of \(0.9 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\)
- Solve the differential equation to show that
$$r ^ { \frac { 3 } { 2 } } = 5.4 t + 10$$ - Hence find the radius of the balloon when \(t = 20\)
Give your answer to the nearest millimetre.
- Suggest a limitation of the model.