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Differential Equations - Modelling with Differential Equations

Differential Equations - Modelling with Differential Equations

IB Year 6 | Grade 12

At time \(t\), the radius of a spherical balloon is \(r\) cm. The balloon is being inflated so that the rate of increase of its radius is inversely proportional to the square root of its radius. When \(t=1\), \(r=4\), and at this instant, the radius is increasing at \(1.5\) cm \(\mathrm{s}^{-1}\).

  1. Form a differential equation and solve for \(r\) in terms of \(t\).
  2. Determine how much air was in the balloon initially?

Solution:

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Answer:(a) \(r=\left(\frac{9t+7}{2}\right)^{\frac{2}{3}}\), \(t\ge0\). (b) \(\frac{49\pi}{3}\text{ cm}^3\approx51.3\text{ cm}^3\).

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