Exponential Growth & Decay (Cambridge International IGCSE Mathematics)

Exponential growth & decay

What is exponential growth?

• When a quantity grows exponentially it is increasing from an original amount by a percentage each year for $n$ 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 percentage each year for $n$ 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 can I model a scenario as exponential growth or decay?

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

• A useful format for this equation is

  • $B=A\times k^{n}$ where:

    • $A$ is the starting (initial) amount

    • $B$ is the new amount

    • $k$ is the appropriate multiplier or scale factor for the growth or decay in the time period

      • E.g. $k=0.8$ for a 20% decay, $k=1.2$ for a 20% growth

    • $n$ is the number of time periods

  • Note if $k>1$ then it is exponential growth

    • If $0<k<1$ then it is exponential decay
    • $k$ cannot be negative

How do I use the exponential growth & decay equation?

• You may need to rearrange the equation $B=A\times k^{n}$

  • To find $A$ giving $A=\frac{B}{k^{n}}$

  • To find $k$ giving $k^{n}=\frac{B}{A}$ so k=n-th root of B / A)

  • To find $n$, using trial and improvement

    • Test different whole-number values for $n$ until both sides of the equation balance

How does exponential growth and decay relate to exponential graphs?

• Plotting the exponential model $B=A\times k^{n}$ on a graph where:

  • $n$ is on the x-axis

  • and $B$ is on the y-axis

  • gives the shape of an exponential graph

    • often written as $y=ak^{x}$

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 the question may require you to round to the nearest integer for $n$

Worked Example An island has a population of 25 000 people.

The population increases exponentially by 4% every year.

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

Answer:

The question says “increases exponentially” so use $B=A\times k^{n}$ where $k>1$

$k$ comes from a percentage increase so add 0.04 to 1

$k=1+0.04$

Substitute $A$ = 25 000, $k$ = 1.04 and $n$ = 13 into the formula

$25000\times1.04^{13}$* *

Work out the value on your calculator

41626.83…

Round to the nearest hundred

Final answer: 41 600 people

Worked Example The temperature of a cup of coffee exponentially decays from 60°C by $r$% each hour. After 3 hours, the temperature is 18°C.

Find the value of $r$ to 3 significant figures.

Answer:

The question says “exponentially decays” so use $B=A\times k^{n}$ where $0<k<1$ Note that $k$ is the multiplier (it is not equal to $r$ in the question, but is related) Substitute $A$ = 60 and $n$ = 3 into the equation

$60\times k^{3}$

The temperature after 3 hours is 18, so set the whole equation equal to 18

$60\times k^{3}=18$

Solve this equation for $k$ Start by dividing both sides by 60

$k^{3}=0.3$

The left hand side is to the power of 3 (cubed) So cube-root both sides and write out lots of decimal places

k=∛(0.3)=0.669432950...

Find the percentage decrease represented by this number It may help to think of an example, e.g. $k$ = 0.6 represents a decrease of 40%

$1-0.669432950...=0.3305670499...$

It represents a decrease by 33.05670...% Round to 3 significant figures

Final answer: $r$= 33.1