Higher November 2023 Paper 4 Q16
16 A biologist assumes the population, \(P\), of birds on an island can be predicted using the formula
\[P = 3800 \times 1.042^n\]where \(n\) is the number of years after the start of 2020.
(a) Write down the percentage increase per year that is used in the formula. [1]
(b) Calculate the predicted population at the start of 2024. [2]
(c)
(i) Show that the number of birds is predicted to exceed 7000 during 2034. [3]
(ii) A researcher says that between 2022 and 2030 the percentage increase per year in the population will be 2.8%.
If the researcher is correct, explain how this new information will affect the answer in part (c)(i). [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4.2 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4479 to 4480 | 2 | M1 for \(3800 \times 1.042^4\) oe | condone answer 4500 with M1 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) [2034=] 6759 to 6760 and [2035=] 7043 to 7044 | 3 | B2 for 6759 to 6760 or 7043 to 7044 or M1 for \(3800 \times 1.042^{14}\) oe or \(3800 \times 1.042^{15}\) oe | Also condone values of n between 14 and 15, for 3 marks one must be below 7000 and one at or above 7000, each rot to at least the nearest integer (see appendix) |
| (ii) It will take longer than 2034 [ for the population to reach 7000] | 1 | their response must relate to the figures in (c)(i) e.g condone “decrease the number in part (c)(i)” | |
Appendix: Question 16(c)(i)
| 14.1 | 6787.63 |
| 14.2 | 6815.62 |
| 14.3 | 6843.72 |
| 14.4 | 6871.93 |
| 14.5 | 6900.26 |
| 14.6 | 6928.71 |
| 14.7 | 6957.27 |
| 14.8 | 6985.96 |
| 14.9 | 7014.76 |
Other numbers will need calculating.
Logarithms
\(3800(1.042)^n = 7000\)
\((1.042)^n = 7000 \div 3800 = 1.842\ldots\)
\(n = \log 1.842 \div \log 1.042\)
\(n = 14.8[48\ldots]\) or 14.85 will score M2
2020 + 14.85 [= 2034.8…] or 2034 – 2020 [= 14] scores A1
Appendix: exemplar responses for Q16(c)(ii)
| Response | Mark |
|---|---|
| It will take longer to reach 7000 | 1 |
| It will not exceed 7000 by 2034 or 2035 | 1 |
| the population will be lower in 2034 | 1 |
| the population growth will be slower | 1 |
| there will be less birds [than expected] | 1 |
| they will need to use a different equation to calculate the population | 0 |
| the answer will reduce (or increase) | 0 |
| It will have a different outcome | 0 |
| The population of birds is going to decrease | 0 |