11 David puts a block of ice into a cool-box. He wishes to model the mass \(m \mathrm {~kg}\) of the remaining block of ice at time \(t\) hours later. He finds that when \(t = 5 , m = 2.1\), and when \(t = 50 , m = 0.21\).
- David at first guesses that the mass may be inversely proportional to time. Show that this model fits his measurements.
- Explain why this model
- is not suitable for small values of \(t\),
- cannot be used to find the time for the block to melt completely.
David instead proposes a linear model \(m = a t + b\), where \(a\) and \(b\) are constants.
- Find the values of the constants for which the model fits the mass of the block when \(t = 5\) and \(t = 50\).
- Interpret these values of \(a\) and \(b\).
- Find the time according to this model for the block of ice to melt completely.