Logarithmic graphs

Show how

– kt θ = 21 – A e

can be rearranged to obtain

ln (21 – θ) = – kt + ln A

The researcher uses measurements they have recorded to plot the graph of ln(21 – θ) against t as shown in the diagram below.

Use the information on the graph to find the value of A

Give your answer to three significant figures.

Use the information on the graph to find the value of k

Give your answer to three significant figures.

Find the temperature of the dessert when it is initially removed from the freezer.

Give your answer to three significant figures.

The dessert is ready to be eaten when its temperature reaches 4 °C

Use the model to determine the time, after being removed from the freezer, for the dessert to reach this temperature.

Give your answer to the nearest 10 minutes.

Victoria, a market researcher, believes the average weekly value, £ V million, of online grocery sales in the UK has grown exponentially since 2009.

Victoria models the incomplete data, shown in the table, using the formula

where N is the number of years since 2009 and a and b are constants.

Victoria wishes to determine the values of a and b in her formula.

To do this she plots a graph of log₁₀ V against N and then draws a line of best fit as shown in the diagram below.

The equation of Victoria’s line of best fit is

Use the equation of Victoria’s line of best fit to show that, correct to three significant figures, a = 57.5

Use the equation of Victoria’s line of best fit to find the value of b

Give your answer to three significant figures.

According to Victoria’s model, state the yearly percentage increase in the average weekly value of online grocery sales.

Use Victoria’s model to predict the average weekly value of online grocery sales in 2025.

Explain why the prediction made in part (c)(i) may be unreliable.

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 log₁₀ T against log₁₀ d is shown with data for Mercury and Uranus labelled.

Find the equation of the straight line in the form

where a and b are constants to be found.

Show that

where K and n are constants to be found.

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.

The table below shows the annual global production of plastics, P, measured in millions of tonnes per year, for six selected years.

Year	1980	1985	1990	1995	2000	2005
	75	94	120	156	206	260

It is thought that P can be modelled by

where t is the number of years after 1980 and A and k are constants.

Show algebraically that the graph of log₁₀ P against t should be linear.

Complete the table below.

	0	5	10	15	20	25
log10	1.88	1.97	2.08		2.31

Plot log₁₀ P against t, and draw a line of best fit for the data.

Hence, show that k is approximately 0.02

Find the value of A.

Using the model with k 002 predict the number of tonnes of annual global production of plastics in 2030.

Using the model with k 002 predict the year in which P first exceeds 8000

Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics.

Theresa bought a house on 2 January 1970 for £8000.

The house was valued by a local estate agent on the same date every 10 years up to 2010.

The valuations are shown in the following table.

The valuation price of the house can be modelled by the equation

where V pounds is the valuation price t years after 2 January 1970 and p and q are constants.

Show that V pq^t can be written as log₁₀ V log₁₀ p t log₁₀ q

The values in the table of log 10 V against t have been plotted and a line of best fit has been drawn on the graph below.

Using the given line of best fit, find estimates for the values of p and q.

Give your answers correct to three significant figures.

Determine the year in which Theresa’s house will first be worth half a million pounds.

Explain whether your answer to part (c) is likely to be reliable.