Exponential growth and decay

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The value of a car, £V, is modelled by the equation

where A and k are positive constants and t is the age of the car in years. Given that

• the initial value of the car was £20 000

• the value of the car was £12 000 when it was 2.5 years old

find a complete equation for the model, giving the exact value of A and the value of k to 3 significant figures.

Show that the rate of change in the value of the car can be expressed in the form – k (V – 1500)

State a limitation of this model.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A new type of car is released for sale. The total number of this type of car sold, N, in a particular region, t months after the cars were released for sale, is modelled by the equation

Use the equation of the model to answer parts (a), (b), (c) and (d).

Find the total number of cars sold in the first 3 months. Given that N = 3000 when t = T

find the value of T giving the answer to 2 decimal places.

Find the rate of increase in the total number of cars sold when t = 3, giving the answer to 3 significant figures. After a marketing campaign, the total number of cars sold is expected to rise and have an upper limit of 6500

Using this information, suggest one refinement to the model.

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.

Coffee is poured into a cup. The temperature of the coffee, H°C, t minutes after being poured into the cup is modelled by the equation

where A and B are constants. Initially, the temperature of the coffee was 85 °C.

State the value of A. Initially, the coffee was cooling at a rate of 7.5 °C per minute.

Find a complete equation linking H and t, giving the value of B to 3 decimal places.

A scientist is studying the growth of two different populations of bacteria. The number of bacteria, N, in the first population is modelled by the equation

where A and k are positive constants and t is the time in hours from the start of the study. Given that

• there were 1000 bacteria in this population at the start of the study

• it took exactly 5 hours from the start of the study for this population to double

find a complete equation for the model.

Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures. The number of bacteria, M, in the second population is modelled by the equation

where k has the value found in part (a) and t is the time in hours from the start of the study. Given that T hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of T.

A scientist is studying the growth of two different populations of bacteria.

The number of bacteria, N, in the first population is modelled by the equation

where A and k are positive constants and t is the time in hours from the start of the study.

Given that

• there were 1000 bacteria in this population at the start of the study

• it took exactly 5 hours from the start of the study for this population to double

find a complete equation for the model.

Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, M, in the second population is modelled by the equation

where k has the value found in part (a) and t is the time in hours from the start of the study.

Given that T hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of T.

A quantity of ethanol was heated until it reached boiling point. The temperature of the ethanol, θ°C, at time t seconds after heating began, is modelled by the equation

where A and B are positive constants. Given that ● the initial temperature of the ethanol was 18 °C ● after 10 seconds the temperature of the ethanol was 44 °C

find a complete equation for the model, giving the values of A and B to 3 significant figures. Ethanol has a boiling point of approximately 78 °C

Use this information to evaluate the model.

In a simple model, the value, £ V, of a car depends on its age, t, in years.

The following information is available for car A

its value when new is £20 000 its value after one year is £16 000

Use an exponential model to form, for car A, a possible equation linking V with t.

The value of car A is monitored over a 10-year period. Its value after 10 years is £2 000

Evaluate the reliability of your model in light of this information.

The following information is available for car B

it has the same value, when new, as car A its value depreciates more slowly than that of car A

Explain how you would adapt the equation found in (a) so that it could be used to model the value of car B.

The value, £ V, of a vintage car t years after it was first valued on 1st January 2001, is modelled by the equation

V=Ap^twhere A and p are constants

Given that the value of the car was £32 000 on 1st January 2005 and £50 000 on 1st January 2012

find p to 4 decimal places,

show that A is approximately 24 800

With reference to the model, interpret

the value of the constant A,

the value of the constant p.

Using the model,

find the year during which the value of the car first exceeds £100 000

A scientist is studying a population of mice on an island.

The number of mice, N, in the population, t months after the start of the study, is modelled by the equation

Find the number of mice in the population at the start of the study.

The rate of growth is a maximum after T months.

Find, according to the model, the value of T.

According to the model, the maximum number of mice on the island is P.

State the value of P.

The current, I amps, in an electric circuit at time t seconds is given by