6 Victoria, a market researcher, believes the average weekly value, \(\pounds V\) million, of online grocery sales in the UK has grown exponentially since 2009.
Victoria models the incomplete data, shown in the table, using the formula
$$V = a \times b ^ { N }$$
where \(N\) is the number of years since 2009 and \(a\) and \(b\) are constants.
| Year | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 |
| Average Weekly Sales | | \(\pounds V\) million |
| 56.4 | | 74.5 | 86.9 | 97.7 | 109.3 | | 141.9 |
6
- Victoria wishes to determine the values of \(a\) and \(b\) in her formula.
To do this she plots a graph of \(\log _ { 10 } V\) against \(N\) and then draws a line of best fit as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-08_757_1040_1169_589}
The equation of Victoria's line of best fit is
$$\log _ { 10 } V = 0.057 N + 1.76$$
6 - Use the equation of Victoria's line of best fit to show that, correct to three significant figures, \(a = 57.5\)
[0pt]
[1 mark]
6
- (ii) Use the equation of Victoria's line of best fit to find the value of \(b\)
Give your answer to three significant figures.
6 - According to Victoria's model, state the yearly percentage increase in the average weekly value of online grocery sales.
6
- Use Victoria's model to predict the average weekly value of online grocery sales in 2025.
6
- (ii) Explain why the prediction made in part (c)(i) may be unreliable.