How to Solve Exponential Growth and Decay Word Problems
Exponential growth and decay are fundamental concepts in mathematics that are widely used in various real-world scenarios. Whether it’s population growth, radioactive decay, or compound interest, understanding how to solve exponential growth and decay word problems is essential. In this article, we will discuss the steps and techniques to solve these types of problems effectively.
Understanding the Problem
The first step in solving exponential growth and decay word problems is to understand the problem thoroughly. Identify the given information, such as the initial value, growth or decay rate, and the time period. Make sure you understand whether the problem involves exponential growth or decay.
Identifying the Formula
Once you have a clear understanding of the problem, the next step is to identify the appropriate formula. For exponential growth, the formula is:
\[ P(t) = P_0 \times (1 + r)^t \]
where \( P(t) \) is the value at time \( t \), \( P_0 \) is the initial value, \( r \) is the growth rate, and \( t \) is the time period.
For exponential decay, the formula is:
\[ P(t) = P_0 \times (1 – r)^t \]
where \( P(t) \) is the value at time \( t \), \( P_0 \) is the initial value, \( r \) is the decay rate, and \( t \) is the time period.
Converting Percentages to Decimal Form
In many word problems, the growth or decay rate is given as a percentage. To use the formulas, you need to convert these percentages to decimal form. To do this, divide the percentage by 100. For example, a 5% growth rate would be 0.05 in decimal form.
Solving the Problem
Now that you have the formula and the decimal form of the growth or decay rate, you can solve the problem. Substitute the given values into the formula and calculate the result. For example, if you are given an initial value of 100, a growth rate of 5%, and a time period of 3 years, the calculation would be:
\[ P(t) = 100 \times (1 + 0.05)^3 \]
\[ P(t) = 100 \times 1.05^3 \]
\[ P(t) = 100 \times 1.157625 \]
\[ P(t) = 115.7625 \]
So, after 3 years, the value would be 115.7625.
Checking Your Answer
After solving the problem, it’s essential to check your answer. Make sure it makes sense in the context of the problem. For example, if you are solving a population growth problem, your answer should be a positive number, as population cannot be negative.
Conclusion
Solving exponential growth and decay word problems requires a clear understanding of the problem, the appropriate formula, and the conversion of percentages to decimal form. By following these steps and techniques, you can effectively solve these types of problems and apply them to real-world scenarios.
