1.13.2 Exponential decay
When a quantity exponentially decays, it decreases by \( r \)% per year for \( t \) years from an initial amount, \( P \).
Real-world examples of exponential decay include hot water cooling, the value of a car decreasing over time, and radioactive decay
The same formula from depreciation is used
\( A=P(1- \frac{r}{100})^t \)
Where
\( A \)= final amount of the quantity
\( P \)= principal amount of the quantity
\( r \)= percentage decrease
\( t \)= time period
Worked example:
The population of a village decreases exponentially at a rate of \( 4 \)% each year.
The population is now \( 255 \).
Calculate the population \( 16\) years ago.
Substitute \( A=255 \) , \( r=4 \) and \( t=16 \) in the formula \( A=P(1- \frac{r}{100})^t \) to find \( P \).
\( 255=P(1- \frac{4}{100})^{16} \)
Rearrange to make \( P \) the subject
\( P= \frac{255}{{(1- \frac{4}{100})}^{16}} \)
\( P=490.0049325 \)
The answer is a number of people, so we need to round the answer to the nearest whole number.
\( 490 \)
The population was \( 490\).
Test yourself
Question 1:
The population of a city is increasing exponentially at a rate of \( 2 \)% each year.
The population now is \( 256000 \).
Calculate the population after \( 30 \) years.Give your answer correct to the nearest thousand.
[3]
Question 2:
Over a period of \( 3 \) years, a company’s sales of biscuits increased from \( 15.6 \) million packets
to \( 20.8 \) million packets.
The sales increased exponentially by the same percentage each year.
Calculate the percentage increase each year.
[3]
Question 3:
Mohsin’s earnings increase exponentially at a rate of \( 8.7 \)% each year.
During \( 2018 \) he earned \( $ \ 195600 \).
During \( 2027 \), how much more does he earn than during \( 2018 \)?
[3]
Question 4:
In a city the population is increasing exponentially at a rate of \( 1.6 \)% per year.
Find the overall percentage increase at the end of \( 20 \) years.
[2]
Question 5:
Over a period of \( 3 \) years, a company’s sales of biscuits increased from \( 15.6 \) million packets to
\( 20.8 \) million packets.
The sales increased exponentially by the same percentage each year.
Calculate the percentage increase each year.
[3]