June 2024 Paper 2 Q8
8 A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula
\[y = a + b\log_{10} x\]where \(y\) is the median mass in kilograms, \(x\) is age in months and \(a\) and \(b\) are constants.
The zookeeper uses the data shown below to determine the values of \(a\) and \(b\).
| Age in months (\(x\)) | 3 | 24 |
|---|---|---|
| Median mass (\(y\)) | 6.4 | 12 |
(a) The zookeeper uses the data for monkeys aged 3 months to write the correct equation\[6.4 = a + b\log_{10} 3\]
(i) Use the data for monkeys aged 24 months to write a second equation. [1 mark]
(ii) Show that\[b = \frac{5.6}{\log_{10} 8}\] [3 marks]
(iii) Find the value of \(a\).
Give your answer to two decimal places. [1 mark]
(b) Use a suitable value for \(x\) to determine whether the model can be used to predict the median mass of monkeys less than one week old. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| (i) Obtains \(12 = a + b\log_{10} 24\) ISW | B1 | 3.4 |
| (1) | ||
| (ii) Eliminates \(a\) to obtain an equation in \(b\) | M1 | 3.1a |
| Obtains \(b\log_{10} h\) from \(b\log_{10}\) their \(24 - b\log_{10} 3\) or \(b\log_{10}\dfrac{\text{their } 24}{3}\) where 3\(h\) = their 24 | M1 | 1.1a |
| Completes a reasoned argument to show \(b = \dfrac{5.6}{\log_{10} 8}\) Must include \(\log_{10}\dfrac{24}{3}\) OE or \(\log_{10} 24 = \log_{10} 8 \times 3\) AG | R1 | 2.1 |
| (3) | ||
| (iii) Obtains AWRT 3.44 | B1 | 1.1b |
| (1) |
Typical solution
(i)
\[12 = a + b\log_{10} 24\](ii)
\[12 = a + b\log_{10} 24\]\[-(6.4 = a + b\log_{10} 3)\]\[5.6 = b\log_{10} 24 - b\log_{10} 3\]\[= b\log_{10}\frac{24}{3}\]\[= b\log_{10} 8\]\[b = \frac{5.6}{\log_{10} 8}\](iii)
\[a = 3.44\]| Scheme | Marks | AO |
|---|---|---|
| Substitutes a value for \(0 \lt x \leqslant 0.25\) into the model with their \(a\) and \(b\) = AWRT 6.2 PI by correct negative \(y\)-value Or Substitutes \(x\) = 0 into the model with their \(a\) and \(b\) = AWRT 6.2 and states that the value for \(y\) is undefined Or Substitutes \(y\) = 0 into the correct model with and gets \(x\) = AWRT 0.28 | M1 | 3.4 |
| Completes reasoned argument to find a correct median mass for their value of \(x\) and concludes that the model cannot be used to predict the median mass of monkeys less than one week old. Condone that the model cannot be used to predict the median mass of monkeys for their value of \(x\) where \(0 \lt x \leqslant 0.25\) Condone omittance of median or use of weight throughout | R1 | 3.5a |
| (2) | ||
| (7 marks) |
Typical solution
When \(x = 0.25\)
\[y = 3.44 + 6.2\log_{10} 0.25\]\[= -0.29\]The model predicts a negative median mass for a monkey that is one week old, therefore it is unsuitable for use with monkeys 1 week old or less.