Continuous Growth
One last type of exponential growth modeling we can look at is the continuous growth model. When \(\$1.00\) is invested at \(100\%\) interest for \(1\) year, with ever increasing compounding periods, what would the ending value approach? Explore this question by entering powers of \(10\) in the table below:
One last type of exponential growth modeling we can look at is the continuous growth model. When \(\$1.00\) is invested at \(100\%\) interest for \(1\) year, with ever increasing compounding periods, what would the ending value approach? Explore this question by entering powers of \(10\) in the table below:
The value approaches \(2.71828...\) and is called the Euler Number, which has been named \(e\). This can be found in two places on the graphing calculator (2nd LN if you want to raise \(e\) to a power or 2nd division sign if you just want \(e\)).
As a result, instead of using the compound interest formula, we can use a new formula for continuous growth which utilizes \(e\) as the base instead of \(\left(1+r\right)\) or \(\left(1+\frac{r}{n}\right)\)
Continuous Growth Formula: \(y=Pe^{rt}\)
In the formula, \(P\) represents the initial quantity (the value at time = \(0\)), \(r\) is the rate of growth (converted to a decimal), \(t\) represents a period of time, and \(y\) represents the ending quantity after a period of time.
In the formula, \(P\) represents the initial quantity (the value at time = \(0\)), \(r\) is the rate of growth (converted to a decimal), \(t\) represents a period of time, and \(y\) represents the ending quantity after a period of time.
Example 1: A population of bacteria grows continuously at a rate of \(3\%\). If the initial sample weighs \(5\) grams, how much will the sample weigh after \(24\) hours?
\(\begin{align}&y=Pe^{rt}\ & \ \ \ \ \ \ &\text{1) Write the general formula.}\\
&P=5\ \ \ r=3\%=.03\ \ \ t=24\ & \ \ \ \ \ \ &\text{2) Identify the values of the variables.}\\
&y=5e^{\left(.03\cdot24\right)}\ & \ \ \ \ \ \ &\text{3) Substitute the values into the formula.}\\
&y=10.27216605\ & \ \ \ \ \ \ &\text{4) Enter the values into the calculator.}\end{align}\)
The sample will weigh approximately \(10.272\) grams after \(24\) hours.
Example 2: What is the value of a \(\$5500\) investment after \(6\) years if it earns $\(3.82\%\) interest compounded continuously. Assume there are no additional deposits or withdrawals.