1.8 Exponentials and Logarithms

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

Question 47

An ecological study tracked the expansion of an invasive water lily population in a lake starting on 1 March. The total surface area covered by the lilies, $S\,$ square metres, was recorded over several months. The area is modelled by the equation

S = ab^t

where $t\,$ is the number of months since 1 March, and $a\,$ and $b\,$ are constants.

Show that $S = ab^t$ can be written in the form $log_{10} S = \log_{10} a + t \log_{10} b$ .

[2]

The values of $log_{10} S$ against $t\,$ are plotted on a graph using data collected for $\leq t \leq 8$ , and a line of best fit is drawn. The line passes through the points $(0, 1.48)$ and $(8, 3.72)$ . Find estimates for the values of $a\,$ and $b$ . Give your answers to three significant figures.

[4]

Use the model to predict the month in which the surface area covered by the lilies will first exceed 25,000 $m^2$ .

[3]

State one reason why the prediction in part (c) may be unreliable.

[1]

1.8 Exponentials and Logarithms

Question 47

S = ab^t

where $t\,$ is the number of months since 1 March, and $a\,$ and $b\,$ are constants.

Show that $S = ab^t$ can be written in the form $log_{10} S = \log_{10} a + t \log_{10} b$ .

[2]

[4]

Use the model to predict the month in which the surface area covered by the lilies will first exceed 25,000 $m^2$ .

[3]

State one reason why the prediction in part (c) may be unreliable.

[1]

1.8 Exponentials and Logarithms

Question 47

Markscheme

1.8 Exponentials and Logarithms

Question 47

Markscheme

1.8 Exponentials and Logarithms

Question 47

1.8 Exponentials and Logarithms Questions

1.8 Exponentials and Logarithms

Question 47

1.8 Exponentials and Logarithms Questions