ln(y) vs ln(x) linear graph

3 \includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-04_714_515_262_804} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { k }\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.56,2.87\) ) and ( \(0.81,3.47\) ), as shown in the diagram. Find the value of \(k\), and the value of \(A\) correct to 2 significant figures.

3 Two variable quantities \(x\) and \(y\) are related by the equation $$y = A x ^ { n }$$ where \(A\) and \(n\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-2_422_697_977_740} When a graph is plotted showing values of \(\ln y\) on the vertical axis and values of \(\ln x\) on the horizontal axis, the points lie on a straight line. This line crosses the vertical axis at the point ( \(0,2.3\) ) and also passes through the point (4.0,1.7), as shown in the diagram. Find the values of \(A\) and \(n\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d90dc270-b304-4b42-8e0e-37641b8a03b8-2_556_1113_680_516} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { m }\), where \(K\) and \(m\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 0,2.0 )\) and \(( 6,10.2 )\), as shown in the diagram. Find the values of \(K\) and \(m\), correct to 2 decimal places.

3 The variables \(x\) and \(y\) satisfy the equation \(x ^ { n } y = C\), where \(n\) and \(C\) are constants. When \(x = 1.10\), \(y = 5.20\), and when \(x = 3.20 , y = 1.05\).

1. Find the values of \(n\) and \(C\).
2. Explain why the graph of \(\ln y\) against \(\ln x\) is a straight line.

3 \includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-05_666_800_260_667} The variables \(x\) and \(y\) satisfy the equation \(x ^ { n } y ^ { 2 } = C\), where \(n\) and \(C\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 0.31,1.21 )\) and \(( 1.06,0.91 )\), as shown in the diagram. Find the value of \(n\) and find the value of \(C\) correct to 2 decimal places.

6 Fig. 6 shows the relationship between \(\log _ { 10 } x\) and \(\log _ { 10 } y\). \begin{figure}[h] \end{figure} Find \(y\) in terms of \(x\).

8 Fig. 8 shows the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\). It is a straight line passing through the points \(( 2,8 )\) and \(( 0,2 )\). \begin{figure}[h] \end{figure} Find the equation relating \(\log _ { 10 } y\) and \(\log _ { 10 } x\) and hence find the equation relating \(y\) and \(x\).

13. \begin{figure}[h] \end{figure} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = p m ^ { q }$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg .
Figure 2 illustrates the linear relationship between \(\log _ { 10 } h\) and \(\log _ { 10 } m\) The line meets the vertical \(\log _ { 10 } h\) axis at 2.25 and has a gradient of - 0.235

1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute.
2. Comment on the suitability of the model for this mammal.
3. With reference to the model, interpret the value of the constant \(p\).

1. 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]{f7994129-07ee-4f6d-9531-08a15a38b794-18_1232_1046_804_513} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 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. Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1000000 .
4. With reference to the model, interpret the value of the constant \(a\).

1. The time, \(T\) seconds, that a pendulum takes to complete one swing is modelled by the formula

$$T = a l ^ { b }$$ where \(l\) metres is the length of the pendulum and \(a\) and \(b\) are constants.

1. Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-26_581_888_749_625} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A student carried out an experiment to find the values of the constants \(a\) and \(b\).
   The student recorded the value of \(T\) for different values of \(l\).
   Figure 3 shows the linear relationship between \(\log _ { 10 } l\) and \(\log _ { 10 } T\) for the student's data.
   The straight line passes through the points \(( - 0.7,0 )\) and \(( 0.21,0.45 )\) Using this information,
2. find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of \(a\) and the value of \(b\), each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(a\).

7 The relationship between the variables \(P\) and \(Q\) is modelled by the formula \(Q = a P ^ { n }\) where \(a\) and \(n\) are constants.
Some values of \(P\) and \(Q\) are obtained from an experiment.

1. State appropriate quantities to plot so that the resulting points lie approximately in a straight line.
2. Explain how to use such a graph to estimate the value of \(n\).

11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h] \end{table} The intensity \(y\) is proportional to a power of the distance \(x\).

1. Write down an equation for \(y\) in terms of \(x\) and two constants.
2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
4. Use the values from part (iii) to find \(a\) and \(b\).
5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
6. Sketch a graph of the equation found in part (v).
7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.

11 A student records the time a pendulum takes to swing for different lengths of pendulum. The student decides to plot a graph of \(\log _ { 10 } T\) against \(\log _ { 10 } l\) where \(T\) is the time in seconds that the pendulum takes to return to its start position and \(l\) is the length in metres of the pendulum. They use a model for \(\log _ { 10 } T\) in terms of \(\log _ { 10 } l\) of the form \(\log _ { 10 } T = \log _ { 10 } \mathrm { k } + \mathrm { n } \log _ { 10 } \mathrm { l }\). The student records the following data points.

1. Determine the values of \(k\) and \(n\) that best model the data. Give your values correct to 2 significant figures.
2. Using these values of \(k\) and \(n\), write the student's model as an equation expressing \(T\) in terms of \(l\).

13 The percentage of the adult population visiting the cinema in Great Britain has tended to increase since the 1980s. The table shows the results of surveys in various years. Source: Department of National Statistics, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
This growth may be modelled by an equation of the form $$P = a t ^ { b } ,$$ where \(P\) is the percentage of the adult population visiting the cinema, \(t\) is the number of years after the year 1985/86 and \(a\) and \(b\) are constants to be determined.

1. Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(\log _ { 10 } t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. \section*{Answer part (ii) of this question on the insert provided.}
2. Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(\log _ { 10 } t\). Draw by eye a line of best fit for the data.
3. Use your graph to find the equation for \(P\) in terms of \(t\).
4. Predict the percentage of the adult population visiting the cinema in the year 2007/2008 (i.e. when \(t = 22\) ), according to this model.

8 Maxine measures the pressure, \(P\) kilopascals, and the volume, \(V\) litres, in a fixed quantity of gas. Maxine believes that the pressure and volume are connected by the equation $$P = c V ^ { d }$$ where \(c\) and \(d\) are constants. Using four experimental results, Maxine plots \(\log _ { 10 } P\) against \(\log _ { 10 } V\), as shown in the graph below. \includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-10_1386_1076_792_482} 8

1. Find the value of \(P\) and the value of \(V\) for the data point labelled \(A\) on the graph.
   8
2. Calculate the value of each of the constants \(c\) and \(d\).
   8
3. Estimate the pressure of the gas when the volume is 2 litres.

10 Raj is investigating how the price, \(P\) pounds, of a brilliant-cut diamond ring is related to the weight, \(C\) carats, of the diamond. He believes that they are connected by a formula $$P = a C ^ { n }$$ where \(a\) and \(n\) are constants.
10

1. Express \(\ln P\) in terms of \(\ln C\).
   10
2. Raj researches the price of three brilliant-cut diamond rings on a website with the following results.

   \(\boldsymbol { C }\)	0.60	1.15	1.50
   \(\boldsymbol { P }\)	495	1200	1720

   10
3. 1. Plot \(\ln P\) against \(\ln C\) for the three rings on the grid below. \includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-15_1018_1467_751_283} 10
4. (ii) Explain which feature of the plot suggests that Raj's belief may be correct.
   10
5. (iii) Using the graph on page 15 , estimate the value of \(a\) and the value of \(n\). 10
6. Explain the significance of \(a\) in this context.
   10
7. Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats. Estimate the price of such a ring.