Model with logarithmic relationship

1. The weight of a baby mammal is monitored over a 16 -month period.

The weight of the mammal, \(w \mathrm {~kg}\), is given by $$w = \log _ { a } ( t + 5 ) - \log _ { a } 4 \quad 2 \leqslant t \leqslant 18$$ where \(t\) is the age of the mammal in months and \(a\) is a constant.
Given that the weight of the mammal was 10 kg when \(t = 3\)

1. show that \(a = 1.072\) correct to 3 decimal places. Using \(a = 1.072\)
2. find an equation for \(t\) in terms of \(w\)
3. find the value of \(t\) when \(w = 15\), giving your answer to 3 significant figures.

8. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted. \(N\) and \(T\) are expected to satisfy a relationship of the form $$N = a T ^ { b } \quad \text { where } a \text { and } b \text { are constants }$$

1. Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-24_1223_1043_895_461} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\)
2. Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
3. With reference to the model, interpret the value of the constant \(a\).

10 A function \(y = \mathrm { f } ( x )\) may be modelled by the equation \(y = a x ^ { b }\).

1. Show why, if this is so, then plotting \(\log y\) against \(\log x\) will produce a straight line graph. Explain how \(a\) and \(b\) may be determined experimentally from the graph.
2. Values of \(x\) and \(y\) are given below. By plotting a graph of logy against log \(x\), show that the model above is appropriate for this set of data and find values of \(a\) and \(b\) given that \(a\) is an integer and \(b\) can be written as a fraction with a denominator less than 10 .

   \(x\)	2	3	4	5	6
   \(y\)	4.6	5.0	5.3	5.5	5.7

3. Use your formula from part (ii) to estimate the value of \(y\) when \(x = 2.8\).

6 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = k x ^ { n }$$ where \(k\) and \(n\) are constants.
Experimental evidence has provided the following approximate values:

1. Complete the table in Figure 1, showing values of \(X\) and \(Y\), where $$X = \log _ { 10 } x \quad \text { and } \quad Y = \log _ { 10 } y$$ Give each value to two decimal places.
2. Show that if \(y = k x ^ { n }\), then \(X\) and \(Y\) must satisfy an equation of the form $$Y = a X + b$$
3. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
4. Find an estimate for the value of \(n\).

4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { b }$$ where \(a\) and \(b\) are constants.

1. Using logarithms to base 10 , reduce the relation \(y = a x ^ { b }\) to a linear law connecting \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
2. The diagram shows the linear graph that results from plotting \(\log _ { 10 } y\) against \(\log _ { 10 } x\). \includegraphics[max width=\textwidth, alt={}, center]{49539feb-f842-49f4-b809-72e8147072e7-3_711_1223_1503_411} Find the values of \(a\) and \(b\).

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 \(b\). 8

1. The zookeeper uses the data for monkeys aged 3 months to write the correct equation $$6.4 = a + b \log _ { 10 } 3$$ 8
2. 1. Use the data for monkeys aged 24 months to write a second equation.
      8
3. (ii) Show that $$b = \frac { 5.6 } { \log _ { 10 } 8 }$$ 8
4. (iii) Find the value of \(a\).
   Give your answer to two decimal places.
   \section*{Question 8 continues on the next question} 8
5. 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.
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-13_2491_1757_173_121}

7

1. 1. Find the equation of the straight line in the form $$\log _ { 10 } T = a + b \log _ { 10 } d$$ where \(a\) and \(b\) are constants to be found.
      7
2. (ii) Show that $$T = \mathrm { K } d ^ { \mathrm { n } }$$ where K and n are constants to be found.
   7
3. Neptune takes approximately 60000 days to complete one orbit of the Sun.
   Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune from the Sun.