13 The thickness of a glacier has been measured every five years from 1960 to 2010. The table shows the reduction in thickness from its measurement in 1960.
| Year | 1965 | 1970 | 1975 | 1980 | 1985 | 1990 | 1995 | 2000 | 2005 | 2010 |
| Number of years since \(1960 ( t )\) | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| Reduction in thickness since \(1960 ( h \mathrm {~m} )\) | 0.7 | 1.0 | 1.7 | 2.3 | 3.6 | 4.7 | 6.0 | 8.2 | 12 | 15.9 |
An exponential model may be used for these data, assuming that the relationship between \(h\) and \(t\) is of the form \(h = a \times 10 ^ { b t }\), where \(a\) and \(b\) are constants to be determined.
- Show that this relationship may be expressed in the form \(\log _ { 10 } h = m t + c\), stating the values of \(m\) and \(c\) in terms of \(a\) and \(b\).
- Complete the table of values in the answer book, giving your answers correct to 2 decimal places, and plot the graph of \(\log _ { 10 } h\) against \(t\), drawing by eye a line of best fit.
- Use your graph to find \(h\) in terms of \(t\) for this model.
- Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model.
- Give one reason why this model will not be suitable in the long term.
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