11 The owners of an online shop believe that their sales can be modelled by \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants, \(S\) is the number of items sold in a month and \(t\) is the number of complete months since starting their online shop. The sales for the first six months are recorded, and the values of \(\log _ { 10 } S\) are plotted against \(t\) in the graph below. The graph is repeated in the Printed Answer Booklet.
\includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-08_1203_1408_552_244}

1. Explain why the graph suggests that the given model is appropriate. The owners believe that \(a = 120\) and \(b = 1.15\) are good estimates for the parameters in the model.
2. Show that the graph supports these estimates for the parameters.
3. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of items sold in the seventh month after opening.
4. 1. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of months after opening when the total number of items sold after opening will first exceed 70000 .
   2. Comment on how reliable this prediction may be.