2026 May TZC P2 Q3

The volume of air inside a balloon, \(V\), measured in cubic metres, after \(t\) minutes is given by

\(V = 5 + 20\mathrm{e}^{-kt}\), where \( t \geq 0\) and \(k\) is a positive constant.

1. Write down the initial volume of air inside the balloon.

   [1]

After \(4\) minutes, the volume of air inside the balloon is half the initial volume.

2. Find the value of \(k\).

   [2]

3. Find the time at which the volume of air inside the balloon is \(8\) cubic metres.

   [2]