Higher June 2025 Paper 4 Q16
16 The population of fish, \(P\), in a pond is predicted by the formula
\[P = 640 \times 1.025^n\]where \(n\) is the number of years after 1st January 2020.
(a) Write down the population of fish on 1st January 2020. [1]
(b) Write down the percentage increase per year used in the formula. [1]
(c)
(i) Show that the formula predicts that the population of fish will reach 1000 between 1st January 2038 and 1st January 2039. [3]
(ii) Give one reason why this might not happen. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 640 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 2.5 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 998 to 999 and 1023 to 1024 | 3 | M2 for one of these correct or M1 for substitution of any number between 18 and 19 into the formula | e.g. \(640 \times 1.025^{18}\) and \(640 \times 1.025^{19}\) or their \(998 \times 1.025\) If they use other values, one value must be 18 and instead of 19 they can use : \(18.08 \leqslant\) other value \(\leqslant 19\) and ignore extras |
| (ii) Any reason suggesting that the rate will not continue | 1 | see appendix, if more than one statement mark the best unless one is contradictory | |
Appendix: exemplar responses for Q16(c)(ii)
| Response | Mark |
|---|---|
| The [percentage] rate does not continue | 1 |
| It will not increase by 2.5% each year | 1 |
| Size of pond is restricted e.g. the more fish means less space so the rate will slow down | 1 |
| Pond cannot support so many fish | 1 |
| There might not be enough food | 1 |
| Catching fish [people, birds, other predators] will reduce the stock | 1 |
| There may be [more] predators | 1 |
| They could get hunted | 1 |
| There may be a [new] disease e.g cause infertility | 1 |
| Pollution may kill more fish than predicted | 1 |
| Conditions in the pond may get worse | 1(BOD) |
| Reproduction rate may decrease | 1 |
| Environmental factors may change such as temperature | 1 |
| Some years may differ | 1(BOD) |
| They might ban fishing so that the rate will increase and it will happen earlier | 1 |
| Other factors might affect breeding | 1(BOD) |
| They might not reproduce [at the same rate] | 1 |
| They may die in large amounts | 1 |
| Population will reach maximum | 1 |
| Population increase may fluctuate | 1 |
| Not all fish will fit [in the pond] | 1 |
| The fish will need to reproduce to keep it going | 0 |
| Some fish might die | 0 |
| It is only a prediction [it may not be accurate] | 0 |
| Population will decrease one year | 0 |
| They might not reproduce at a constant rate | 0 |
| The population might decrease | 0 |
| There are other factors | 0 |