Exponential Growth & Decay

What is exponential growth?

When a quantity grows exponentially it is increasing from an original amount, P, by r % 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
- Compound interest
- Population increases
- Bacterial growth
- Number of people infected by a virus

What is exponential decay?

When a quantity exponentially decays it is decreasing from an original amount, P, by r % 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
- Depreciation
- The temperature of hot water cooling down
- The value of a car decreasing over time
- Radioactive decay (how radioactive a substance is over time)

Examiner Tip

Look out for how the question wants you to give your final answer.
- E.g., It may want the final amount to the nearest thousand.

Worked example

(a)

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.

The question says “increases exponentially”
The population increases by 4% each year, so the multiplier is 1.04
The time period is 13 years, so the multiplier is applied 13 times, 1.04¹³

Therefore the final value after 13 years will be

$25 000 \times 1 . 04^{13}$

Work out this value on your calculator

41626.83…

Round this value to the nearest hundred

41 600 people

(b)

The temperature of a cup of coffee exponentially decays from 60°C by 32% each hour.
What is the temperature of the cup of coffee after 3 hours? Round your answer to 1 decimal place.

The question says “exponentially decays”

The temperature decreases by 32% each hour, so the multiplier is 1 - 0.32 = 0.0.68
The time period is 3 hours, so the multiplier is applied 3 times, 0.68³

Therefore the final value after 3 hours will be

$60 \times 0 . 68^{3}$

Work out this value on your calculator

18.86592

Round to 1 decimal place

$18.9 ° C (1 d . p .)$

Exponential Growth & Decay (Edexcel GCSE Maths: Foundation)

Revision Note