9 The table below shows the annual global production of plastics, \(P\), measured in millions of tonnes per year, for six selected years.
| Year | 1980 | 1985 | 1990 | 1995 | 2000 | 2005 |
| \(\boldsymbol { P }\) | 75 | 94 | 120 | 156 | 206 | 260 |
It is thought that \(P\) can be modelled by
$$P = A \times 10 ^ { k t }$$
where \(t\) is the number of years after 1980 and \(A\) and \(k\) are constants.
9
- Show algebraically that the graph of \(\log _ { 10 } P\) against \(t\) should be linear.
9 - Complete the table below.
| \(\boldsymbol { t }\) | 0 | 5 | 10 | 15 | 20 | 25 |
| \(\boldsymbol { \operatorname { l o g } } _ { \mathbf { 1 0 } } \boldsymbol { P }\) | 1.88 | 1.97 | 2.08 | | 2.31 | |
- (ii) Plot \(\log _ { 10 } P\) against \(t\), and draw a line of best fit for the data.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-13_1203_1308_360_367}
9 - Hence, show that \(k\) is approximately 0.02
9
- (ii) Find the value of \(A\).
9 - Using the model with \(k = 0.02\) predict the number of tonnes of annual global production of plastics in 2030.
9
- Using the model with \(k = 0.02\) predict the year in which \(P\) first exceeds 8000
9 - Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-15_2488_1716_219_153}