Logarithmic graphs

The world human population, P billions, is modelled by the equation

where a and b are constants and t is the number of years after 2004

Using the estimated population figures for the years from 2004 to 2007, a graph is plotted of log₁₀P against t. The points lie approximately on a straight line with

• gradient 0.0054

• intercept 0.81 on the log₁₀P axis

Estimate, to 3 decimal places, the value of a and the value of b.

In the context of the model,

interpret the value of the constant a,

interpret the value of the constant b.

Use the model to estimate the world human population in 2030

Comment on the reliability of the answer to part (c).

The value, V pounds, of a mobile phone, t months after it was bought, is modelled by

where a and b are constants.

Figure 2 shows the linear relationship between log₁₀V and t. The line passes through the points (0, 3) and (10, 2.79) as shown.

Using these points,

find the initial value of the phone,

find a complete equation for V in terms of t, giving the exact value of a and giving the value of b to 3 significant figures.

Exactly 2 years after it was bought, the value of the phone was £320

Use this information to evaluate the reliability of the model.

The time, T seconds, that a pendulum takes to complete one swing is modelled by the formula

where l metres is the length of the pendulum and a and b are constants.

Show that this relationship can be written in the form

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 log₁₀ l and log₁₀ T for the student’s data. The straight line passes through the points (– 0.7, 0) and (0.21, 0.45) Using this information,

find a complete equation for the model in the form

giving the value of a and the value of b, each to 3 significant figures.

With reference to the model, interpret the value of the constant a.

The time, T seconds, that a pendulum takes to complete one swing is modelled by the formula

where l metres is the length of the pendulum and a and b are constants.

Show that this relationship can be written in the form

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 log₁₀ l and log₁₀ T for the student’s data. The straight line passes through the points (– 0.7, 0) and (0.21, 0.45)

Using this information,

find a complete equation for the model in the form

giving the value of a and the value of b, each to 3 significant figures.

With reference to the model, interpret the value of the constant a.

A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions. The engineer measures and records the braking distance, d metres, when the brakes are applied from a speed of V ௘NP Kí. Graphs of d against V and log₁₀d against log₁₀ V were plotted. The results are shown below together with a data point from each graph.

Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula

with k 0.017

Using the information given in Figure 5, with k = 0.017

find a complete equation for the model giving the value of n to 3 significant figures.

Sean is driving this car at 60 km h in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes.

Use your formula to find out if Sean will be able to stop before reaching the puddle.