June 2021 Paper 1 Q7
7 Scientists observed a colony of seabirds over a period of 10 years starting in 2010.
They concluded that the number of birds in the colony, its population \(P\), could be modelled by a formula of the form\[P = a(10^{bt})\]where \(t\) is the time in years after 2010, and \(a\) and \(b\) are constants.
(a) Explain what the value of \(a\) represents. [1 mark]
(b) Show that \(\log_{10} P = bt + \log_{10} a\) [2 marks]
(c) The table below contains some data collected by the scientists.
| Year | 2013 | 2015 |
|---|---|---|
| \(t\) | 3 | |
| \(P\) | 10 200 | 12 800 |
| \(\log_{10} P\) | 4.0086 |
(i) Complete the table, giving the \(\log_{10} P\) value to 5 significant figures. [1 mark]
(ii) Use the data to calculate the value of \(a\) and the value of \(b\). [4 marks]
(iii) Use the model to estimate the population of the colony in 2024. [2 marks]
(d)
(i) State an assumption that must be made in using the model to estimate the population of the colony in 2024. [1 mark]
(ii) Hence comment, with a reason, on the reliability of your estimate made in part (c)(iii). [1 mark]
| Scheme | Marks | AO |
|---|---|---|
| Explains that \(a\) represents the initial population. OE | E1 | 2.4 |
| (1) |
Typical solution
\(a\) is the population in 2010
| Scheme | Marks | AO |
|---|---|---|
| Takes logarithms to base 10 of both sides | M1 | 1.1a |
| Completes derivation convincingly AG | R1 | 2.1 |
| (2) |
Typical solution
\[P = a(10^{bt})\]\[\log_{10} P = \log_{10}(a10^{bt})\]\[= \log_{10} a + \log_{10}(10^{bt})\]\[= \log_{10} a + bt\]| Scheme | Marks | AO |
|---|---|---|
| (i) Completes table correctly, figures seen in table or text or used in part (c)(ii) | B1 | 3.3 |
| (1) | ||
| (ii) Uses data to set up a pair of simultaneous equations | M1 | 3.1a |
| Solves equations for either \(b\) or \(\log_{10} a\) correct | A1 | 1.1b |
| Converts \(\log_{10} a\) to obtain a value of \(a\) or uses their \(b\) and data to calculate \(a\) | M1 | 1.1a |
| Obtains both \(b\) and \(a\) correct. AWRT 0.049 and AWFW 7200 to 7300 | A1 | 1.1b |
| (4) | ||
| (iii) Substitutes their values of \(a\) and \(b\) into model and \(t = 14\) | M1 | 3.4 |
| Calculates correct value of population AWFW 35500 to 35600 FT provided >12800 | A1F | 1.1b |
| (2) |
Typical solution
(i)
| Year | 2013 | 2015 |
| \(t\) | 3 | 5 |
| \(P\) | 10 200 | 12 800 |
| \(\log_{10} P\) | 4.0086 | 4.1072 |
(ii)
\[4.0086 = 3b + \log_{10} a\]\[4.1072 = 5b + \log_{10} a\]\[b = 0.0493\]\[\log_{10} a = 3.8607\]\[a = 7256\](iii)
\[7256 \times 10^{(14 \times 0.0493)}\]\[= 35555\]| Scheme | Marks | AO |
|---|---|---|
| (i) States an appropriate assumption about the model. | E1 | 3.5b |
| (1) | ||
| (ii) Makes appropriate comment about limited data, or length of extrapolation, changing food supply, disease or equivalent specific factor. | E1 | 3.5a |
| (1) | ||
| (12 marks) |
Typical solution
(i)
The value of constant \(b\) does not change after 2020
(ii)
Not very reliable, because it is only based on data from two years