1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form

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

1. The zookeeper uses the data for monkeys aged 3 months to write the correct equation $$6.4 = a + b \log_{10} 3$$
   1. Use the data for monkeys aged 24 months to write a second equation. [1 mark]
   2. Show that $$b = \frac{5.6}{\log_{10} 8}$$ [3 marks]
   3. Find the value of \(a\). Give your answer to two decimal places. [1 mark]
2. 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]

A planet takes \(T\) days to complete one orbit of the Sun. \(T\) is known to be related to the planet's average distance \(d\), in millions of kilometres, from the Sun. A graph of \(\log_{10} T\) against \(\log_{10} d\) is shown with data for Mercury and Uranus labelled. \includegraphics{figure_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. [3 marks]
   2. Show that $$T = K d^n$$ where K and n are constants to be found. [2 marks]
2. Neptune takes approximately 60 000 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. [2 marks]

\includegraphics{figure_2} A town's population, \(P\), is modelled by the equation \(P = ab^t\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded. The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log_{10} P\) for the population over a period of 100 years. The line \(l\) meets the vertical axis at \((0, 5)\) as shown. The gradient of \(l\) is \(\frac{1}{200}\).

1. Write down an equation for \(l\). [2]
2. Find the value of \(a\) and the value of \(b\). [4]
3. With reference to the model interpret
   1. the value of the constant \(a\),
   2. the value of the constant \(b\).
   [2]
4. Find
   1. the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
   2. the number of years it takes the population to reach \(200\,000\), according to the model.
   [3]
5. State two reasons why this may not be a realistic population model. [2]

\includegraphics{figure_4} The value of a sculpture, \(£V\), is modelled by the equation \(V = Ap^t\), where \(A\) and \(p\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1960. The line \(l\) shown in Figure 4 illustrates the linear relationship between \(t\) and \(\log_{10}V\) for \(t \geq 0\). The line \(l\) passes through the point \((0, \log_{10}20)\) and \((50, \log_{10}2000)\).

1. Write down the equation of the line \(l\). [3]
2. Using your answer to part a or otherwise, find the values of \(A\) and \(p\). [4]
3. With reference to the model, interpret the values of the constant \(A\) and \(p\). [2]
4. Use your model, to predict the value of the sculpture, on 1st January 2020, giving your answer to the nearest pounds. [1]

Two quantities are related by the equation \(Q = 1.25P^3\). Explain why the graph of \(\log_{10} Q\) against \(\log_{10} P\) is a straight line. State the gradient of the straight line and the intercept on the \(\log_{10} Q\) axis of the graph. [4]

\includegraphics{figure_1} Red squirrels were introduced into a large wood in Northumberland on 1st June 1996. Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. The scientists found that over time the number of red squirrels can be modelled by the formula $$P = ab^t$$ where \(a\) and \(b\) are constants. The line \(l\), shown in Figure 1, illustrates the linear relationship between \(\log_{10} P\) and \(t\) over a period of 20 years. Using the information given on the graph and using the model, find a complete equation for the model giving the value of \(b\) to 4 significant figures. [4]

\includegraphics{figure_2} A town's population, \(P\), is modelled by the equation \(P = ab^t\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded. The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log_{10} P\) for the population over a period of 100 years. The line \(l\) meets the vertical axis at \((0, 5)\) as shown. The gradient of \(l\) is \(\frac{1}{200}\).

1. Write down an equation for \(l\). [2]
2. Find the value of \(a\) and the value of \(b\). [4]
3. With reference to the model interpret
   1. the value of the constant \(a\),
   2. the value of the constant \(b\).
   [2]
4. Find
   1. the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
   2. the number of years it takes the population to reach 200000, according to the model.
   [3]
5. State two reasons why this may not be a realistic population model. [2]

\includegraphics{figure_2} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = pm^q$$ 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 \(q\). [3]

A particular mammal has a mass of 5kg and a resting heart rate of 119 beats per minute.

1. Comment on the suitability of the model for this mammal. [3]
2. With reference to the model, interpret the value of the constant \(p\). [1]

The resting metabolic rate, \(R\) ml of oxygen consumed per hour, of a particular species of mammal is modelled by the formula, $$R = aM^b$$ where • \(M\) grams is the mass of the mammal • \(a\) and \(b\) are constants

1. Show that this relationship can be written in the form $$\log_{10} R = b \log_{10} M + \log_{10} a$$ [2] \includegraphics{figure_3} A student gathers data for \(R\) and \(M\) and plots a graph of \(\log_{10} R\) against \(\log_{10} M\) The graph is a straight line passing through points \((0.7, 1.2)\) and \((1.8, 1.9)\) as shown in Figure 3.
2. Using this information, find a complete equation for the model. Write your answer in the form $$R = aM^b$$ giving the value of each of \(a\) and \(b\) to 3 significant figures. [3]
3. With reference to the model, interpret the value of the constant \(a\) [1]

A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass \(M\) kg of salt remaining after \(t\) minutes by \(M = ak^t\) where \(a\) and \(k\) are constants.

1. Show that the model for \(M\) can be rewritten in the form \(\log_{10} M = t\log_{10} k + \log_{10} a\). [1]

The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.

\(t\) minutes	8	13	21	35	50
\(M\) kg	0.4	0.3	0.2	0.1	0.05

The student uses this data and plots \(y = \log_{10} M\) against \(x = t\) using graph drawing software. The software gives \(y = -0.0214x - 0.2403\) for the equation of the line of best fit.

1. 1. Find the values of \(a\) and \(k\) that follow from the equation of the line. [2]
   2. Interpret the value of \(k\) in context. [1]
2. It is known that when \(t = 0\) the mass of salt in the bucket is 0.5 kg. Comment on the accuracy when the model is used to estimate the initial mass of the salt. [1]
3. Use the model to predict the value of \(t\) at which \(M = 0.01\) kg. [2]
4. Rewrite the model for \(M\) in the form \(M = ae^{-ht}\) where \(h\) is a constant to be determined. [2]

There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows. These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.

1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]

A doctors' surgery starts a campaign to reduce missed appointments. The number of missed appointments for each of the first five weeks after the start of the campaign is shown below.

Number of weeks after the start (\(x\))	1	2	3	4	5
Number of missed appointments (\(y\))	235	149	99	59	38

This data could be modelled by an equation of the form \(y = pq^x\) where \(p\) and \(q\) are constants.

1. Show that this relationship may be expressed in the form \(\log_{10} y = mx + c\), expressing \(m\) and \(c\) in terms of \(p\) and/or \(q\). [2]

The diagram below shows \(\log_{10} y\) plotted against \(x\), for the given data. \includegraphics{figure_5}

1. Estimate the values of \(p\) and \(q\). [3]
2. Use the model to predict when the number of missed appointments will fall below 20. Explain why this answer may not be reliable. [2]

The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4. Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]