13 Answer part (ii) of this question on the insert provided.
The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\).
| Number of months after start-up \(( x )\) | 1 | 2 | 3 | 4 | 5 | 6 |
| Profit for this month \(( \pounds y )\) | 500 | 800 | 1200 | 1900 | 3000 | 4800 |
The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by
$$y = k \times 10 ^ { a x }$$
where \(a\) and \(k\) are constants.
- Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
- On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
- Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
- For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
- State one way in which this model is unrealistic.