Exponential Growth & Decay (Cambridge International IGCSE Maths)

Exponential Growth & Decay

The ideas of compound interest and depreciation can be applied to other (non-money) situations, such as increasing or decreasing populations.

What is exponential growth?

• When a quantity grows exponentially it is increasing from an original amount by a scale factor (or percentage) each year for $x$ years

  • Some questions use a different timescale, such as each day, or each minute

• Real-life examples of exponential growth include:

  • Population increases
  • Bacterial growth
  • The number of people infected by a virus

What is exponential decay?

• When a quantity exponentially decays it is decreasing from an original amount by a scale factor (or percentage) each year for $x$ years

  • Some questions use a different timescale, such as each day, or each minute

• Real-life examples of exponential decay include:

  • The temperature of hot water cooling down
  • The value of a car decreasing over time
  • Radioactive decay (the mass of a radioactive substance over time)

How do I use models for exponential growth or decay?

• Scenarios which exponentially grow or decay can be modelled with an equation

• A useful format for this equation is

  • $y=a^{x}$

    • E.g. $y=4^{x}$ or $y=0.8^{x}$

  • If $a>1$ then it is exponential growth

  • If $0<a<1$ then it is exponential decay

  • $a$ cannot be negative

• $y=3^{x}$ could model the number of flies, $y$, in a population of flies, against the number of days, $x$

  • The initial number of flies, when $x=0$, is $3^{0}=1$
  • After 2 days, the number would be $3^{2}=9$
  • After 4 days, the number would be $3^{4}=81$
  • From the values above, and the fact that $a>1$, this is exponential growth

• $y=0.5^{x}$ could model equationthe radius of a water droplet, $y\mathrm{cm}$, as it evaporates where $x$ is measured in minutes

  • The initial radius of the droplet, when $x=0$, is $0.5^{0}=1\mathrm{cm}$
  • After 2 minutes, the radius would be $0.5^{2}=0.25\mathrm{cm}$
  • After 4 minutes, the radius would be $0.5^{4}=0.0625\mathrm{cm}$
  • From the values above, and the fact that $0<a<1$, this is exponential decay

How are exponential models related to exponential graphs?

• Plotting an exponential model, $y=a^{x}$, on a set of axes gives an exponential graph with the following shape

• The $y$-intercept is always 1

  • This is because the $y$-intercept is where $x=0$
  • $y=a^{0}=1$

• There is an asymptote at equation$y=0$ (the curve gets closer and closer to the x-axis without crossing it)

  • This is because there are no solutions to $0=a^{x}$

Are there any other exponential models I may encounter?

• You may see exponential models in the form $y=ba^{x}$

  • The coefficient, $b$, scales (stretches) the graph vertically

  • The $y$-intercept will be $( 0 , b )$

    • This is because when $x=0$, $y = b a^{0} = b ( 1 ) = b$

  • $b$ is the initial(starting) amount of the quantity being modelled

Exam Hint

• Look out for how the question wants you to give your final answer

  • It may want the final amount to the nearest thousand or to the nearest integer

Worked Example A large population of birds increases exponentially according to the model

$N=1.04^{t}$

where $N$ is the number of birds, measured in thousands, and $t$ is the number of years.

Find the population after 13 years, giving your answer to the nearest hundred.

Substitute in $t=13$

$N=1.04^{13}\,N=1.665073507...$

The question asks for the answer to the nearest hundred The question also states that $N$ is measured in thousands

1.665073507... × 1000 = 1665.073507... birds

Round to nearest hundred

Final answer: 1700

Worked Example The percentage of tickets still available for a concert is modelled approximately by the equation

$y=a^{x}$

Where $y$ is the percentage of tickets still available, as a decimal, and $x$ is the number of days since the tickets have been released.

(a) After 3 days, 51.2% of the tickets are still available. Find the value of equation$a$.

$x$ will be 3 (from 3 days) $y$ will be 0.512 (as the percentage is written as a decimal in this model) Substitute these values into the model

$0.512=a^{3}$

Find $a$ by taking the cube root of both sides

$\begin{gathered} \sqrt[3]{0.512} & = a \\ 0.8 & = a \end{gathered}$

$a=0.8$

(b) What percentage of tickets remain after 2 weeks? State your answer to three significant figures.

Now that $a$ is known, the full equation of the model is

$y=0.8^{x}$

$x$ is measured in days

2 weeks = 2 × 7 = 14 days

Substitute in $x=14$

$y=0.8^{14}=0.04398046511...$

Turn this into a percentage by multiplying by 100

4.398046511... %

Round to three significant figures

Final answer: 4.40 %

(c) State one reason why this model may not be appropriate for the scenario.

Final answer: The value of y can never reach zero (there is an asymptote at$y=0$). The model suggests that the tickets never truly run out (which is unrealistic).