Model y=ab^x: linearise and find constants from graph/data

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. An exponential model may be used for these data, assuming that the relationship between \(h\) and \(t\) is of the form \(h = a \times 10^{bt}\), where \(a\) and \(b\) are constants to be determined.

1. Show that this relationship may be expressed in the form \(\log_{10} h = mt + c\), stating the values of \(m\) and \(c\) in terms of \(a\) and \(b\). [2]
2. 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. [4]
3. Use your graph to find \(h\) in terms of \(t\) for this model. [4]
4. Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model. [2]
5. Give one reason why this model will not be suitable in the long term. [1]

The variables \(y\) and \(x\) are related by an equation of the form $$y = a(b^x)$$ where \(a\) and \(b\) are positive constants. Let \(Y = \log_{10} y\).

1. Show that there is a linear relationship between \(Y\) and \(x\). [2 marks]
2. The graph of \(Y\) against \(x\), shown below, passes through the points \((0, 2.5)\) and \((5, 0.5)\). \includegraphics{figure_3}
   1. Find the gradient of the line. [1 mark]
   2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]

\includegraphics{figure_1} Red squirrels were introduced into a large wood in Northumberland on 1st June 1996. Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. The scientists found that over time the number of red squirrels can be modelled by the formula $$P = ab^t$$ where \(a\) and \(b\) are constants. The line \(l\), shown in Figure 1, illustrates the linear relationship between \(\log_{10} P\) and \(t\) over a period of 20 years. Using the information given on the graph and using the model, find a complete equation for the model giving the value of \(b\) to 4 significant figures. [4]

The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4. Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]