Exponential Growth and Decay Formulas
Exponential Growth: \(y=P\left(1+r\right)^x\)
Exponential Decay: \(y=P\left(1-r\right)^x\)
In both formulas, \(P\) represents the initial quantity (value at time = \(0\)), \(r\) represents the annual rate of growth (this must be converted to a decimal by moving the decimal two places to the left), \(x\) represents the number of growth or decay periods given by the formula \(x=\frac{\text{total time}}{\text{growth or decay time}}\), and \(y\) represents the ending quantity after \(x\) growth periods.
Exponential Decay: \(y=P\left(1-r\right)^x\)
In both formulas, \(P\) represents the initial quantity (value at time = \(0\)), \(r\) represents the annual rate of growth (this must be converted to a decimal by moving the decimal two places to the left), \(x\) represents the number of growth or decay periods given by the formula \(x=\frac{\text{total time}}{\text{growth or decay time}}\), and \(y\) represents the ending quantity after \(x\) growth periods.
Example 1: Naperville’s population has experienced an average yearly rate of growth of \(0.675\%\). The population of Naperville in \(2016\) was \(141,853\) people. If the growth rate remains constant, predict the population of Naperville in \(2025\).
\(\begin{align}&y=P\left(1+r\right)^x\ & \ \ \ \ &\text{1) Write the general formula}\\
&P=141,853\ \ r=.675\%=.00675\ \ x=2025-2016=9\ & \ \ \ \ &\text{2) Identify the values of the variables}\\
&y=141,853\left(1+.00675\right)^9\ & \ \ \ \ &\text{3) Substitute the values into the formula}\\
&y=150,706.9461\ & \ \ \ \ &\text{4) Enter the values into the calculator}\end{align}\)
Round and write your answer in context: The population of Naperville in \(2025\) will be approximately \(150,707\) people.
Example 2: A population of bacteria doubles every \(8\) hours. If the initial population is \(10\) grams, what will the population be in \(2\) days?
Example 3: A mini-van was purchased new in \(2005\) for \(\$30,000\). In \(2017\), it was worth \(\$6300\). What was the rate of depreciation for the mini van?