How to Do Exponential Growth and Decay Word Problems
Exponential growth and decay are two fundamental concepts in mathematics that describe how quantities change over time. Whether you’re dealing with population growth, radioactive decay, or compound interest, understanding how to solve exponential growth and decay word problems is essential. In this article, we will guide you through the process of solving these types of problems step by step.
Step 1: Identify the type of problem
The first step in solving exponential growth and decay word problems is to determine whether the problem involves exponential growth or decay. Exponential growth problems have a positive growth rate, while exponential decay problems have a negative growth rate.
Step 2: Understand the given information
Next, carefully read the problem and identify the given information. Common information includes the initial value, the growth or decay rate, and the time period. Make sure you understand the units of measurement for each piece of information.
Step 3: Write the exponential function
Based on the given information, write the exponential function that represents the problem. For exponential growth, the function will be in the form of f(t) = a(1 + r)^t, where a is the initial value, r is the growth rate, and t is the time. For exponential decay, the function will be in the form of f(t) = a(1 – r)^t, where a is the initial value, r is the decay rate, and t is the time.
Step 4: Solve for the unknown variable
Once you have the exponential function, solve for the unknown variable. This may involve isolating the variable on one side of the equation and using algebraic techniques to simplify the expression.
Step 5: Check your answer
After solving the problem, always check your answer by plugging it back into the original equation. Make sure that the answer makes sense in the context of the problem and that the units of measurement are consistent.
Example Problem
A population of bacteria doubles every hour. If the initial population is 100 bacteria, how many bacteria will there be after 5 hours?
Step 1: Identify the type of problem
This is an exponential growth problem since the population is increasing over time.
Step 2: Understand the given information
Initial value (a) = 100 bacteria
Growth rate (r) = 2 (since the population doubles)
Time period (t) = 5 hours
Step 3: Write the exponential function
f(t) = a(1 + r)^t
f(t) = 100(1 + 2)^5
Step 4: Solve for the unknown variable
f(t) = 100(3)^5
f(t) = 100(243)
f(t) = 24,300
Step 5: Check your answer
After 5 hours, there will be 24,300 bacteria. Plugging this value back into the equation, we get:
f(5) = 100(1 + 2)^5
f(5) = 100(3)^5
f(5) = 24,300
The answer is consistent with the problem’s context and units of measurement.
By following these steps, you can effectively solve exponential growth and decay word problems. Practice with various examples will help you become more comfortable with these types of problems and improve your mathematical skills.
