Mastering the Art of Solving Exponential Growth and Decay Equations- A Comprehensive Guide
How to Solve Exponential Growth and Decay Equations
Exponential growth and decay equations are mathematical models that describe situations where a quantity increases or decreases at a constant percentage rate over time. These equations are widely used in various fields, such as finance, biology, and physics. In this article, we will discuss how to solve exponential growth and decay equations step by step.
Understanding the Basic Form
The basic form of an exponential growth equation is:
\[ P(t) = P_0 \times (1 + r)^t \]
where:
– \( P(t) \) is the quantity at time \( t \)
– \( P_0 \) is the initial quantity
– \( r \) is the growth rate (expressed as a decimal)
– \( t \) is the time
Similarly, the basic form of an exponential decay equation is:
\[ P(t) = P_0 \times (1 – r)^t \]
where the only difference is the negative sign in the growth rate.
Step 1: Identify the Given Values
To solve an exponential growth or decay equation, you first need to identify the given values. These include the initial quantity (\( P_0 \)), the growth rate (\( r \)), and the time (\( t \)).
Step 2: Rearrange the Equation
Next, rearrange the equation to isolate the variable you want to solve for. For example, if you want to find the quantity at a specific time, rearrange the equation to solve for \( P(t) \).
For growth:
\[ P(t) = P_0 \times (1 + r)^t \]
For decay:
\[ P(t) = P_0 \times (1 – r)^t \]
Step 3: Substitute the Given Values
Substitute the given values for \( P_0 \), \( r \), and \( t \) into the rearranged equation.
Step 4: Simplify the Equation
Simplify the equation by performing the necessary calculations. This may involve raising a number to a power or multiplying and dividing by the same number.
Step 5: Solve for the Variable
Finally, solve for the variable you are interested in. For example, if you are solving for \( P(t) \), you will have the quantity at the given time.
Example
Suppose you have an investment that grows at a rate of 5% per year, and you want to find the value of the investment after 10 years. The initial investment is $10,000.
Given:
– \( P_0 = 10,000 \)
– \( r = 0.05 \) (5% as a decimal)
– \( t = 10 \) years
Using the growth equation:
\[ P(t) = P_0 \times (1 + r)^t \]
Substitute the given values:
\[ P(10) = 10,000 \times (1 + 0.05)^{10} \]
Simplify the equation:
\[ P(10) = 10,000 \times (1.05)^{10} \]
Calculate the result:
\[ P(10) = 10,000 \times 1.62889462677744 \]
\[ P(10) \approx 16,288.95 \]
So, the value of the investment after 10 years will be approximately $16,288.95.
In conclusion, solving exponential growth and decay equations involves identifying the given values, rearranging the equation, substituting the values, simplifying the equation, and solving for the variable. By following these steps, you can find the quantity at a specific time or determine other characteristics of the exponential growth or decay process.
