Growth And Decay Word Problems

Mastering Growth and Decay Word Problems: A Comprehensive Guide

Growth and decay word problems are a common application of exponential functions in mathematics. Understanding these problems is crucial for various fields, from finance and biology to engineering and computer science. This comprehensive guide will equip you with the tools and strategies to confidently tackle even the most challenging growth and decay scenarios. We'll explore the underlying concepts, delve into various problem types, and provide step-by-step solutions to solidify your understanding. This guide will cover exponential growth, exponential decay, and the crucial role of half-life in decay problems.

Understanding Exponential Growth and Decay

At the heart of growth and decay problems lies the exponential function. The general form of an exponential function is:

A = P(1 + r)^t

Where:

• A represents the final amount or value.
• P represents the initial amount or principal.
• r represents the growth or decay rate (expressed as a decimal). For growth, r is positive; for decay, r is negative.
• t represents the time period.

Exponential Growth: In exponential growth problems, the quantity increases over time at a rate proportional to its current value. Examples include population growth, compound interest, and the spread of certain diseases. The formula for exponential growth is a direct application of the general exponential function with a positive r.

Exponential Decay: In exponential decay problems, the quantity decreases over time at a rate proportional to its current value. Examples include radioactive decay, the depreciation of assets, and the cooling of objects. The formula remains the same, but r will be negative, reflecting the decrease.

Step-by-Step Approach to Solving Growth and Decay Word Problems

Solving growth and decay word problems involves a systematic approach. Here's a step-by-step guide:

1. Identify the type of problem: Is it exponential growth or decay? Carefully read the problem statement to determine whether the quantity is increasing or decreasing.

2. Identify the known variables: Determine the values of P (initial amount), r (rate), and t (time). Make sure the rate is expressed as a decimal.

3. Choose the appropriate formula: Use the general exponential function, adjusting the sign of r based on whether it's growth (+r) or decay (-r).

4. Substitute the known values: Carefully substitute the known values into the chosen formula.

5. Solve for the unknown variable: Use algebraic manipulation to solve for the unknown variable (A, P, r, or t). This might involve logarithms if the unknown variable is in the exponent.

6. Interpret the solution: Ensure your answer makes sense within the context of the problem.

Examples of Growth Problems

Example 1: Compound Interest

Suppose you invest $1000 in a savings account that pays 5% annual interest, compounded annually. How much money will you have in the account after 3 years?

1. Type: Exponential Growth

2. Known Variables: P = $1000, r = 0.05, t = 3

3. Formula: A = P(1 + r)^t

4. Substitution: A = 1000(1 + 0.05)^3

5. Solution: A = 1000(1.05)^3 ≈ $1157.63

6. Interpretation: After 3 years, you will have approximately $1157.63 in your savings account.

Example 2: Population Growth

The population of a city is currently 50,000 and is growing at a rate of 2% per year. What will the population be in 10 years?

1. Type: Exponential Growth

2. Known Variables: P = 50,000, r = 0.02, t = 10

3. Formula: A = P(1 + r)^t

4. Substitution: A = 50000(1 + 0.02)^10

5. Solution: A = 50000(1.02)^10 ≈ 60950

6. Interpretation: The population will be approximately 60,950 in 10 years.

Examples of Decay Problems

Example 3: Radioactive Decay

A radioactive substance has a half-life of 10 years. If you start with 100 grams of the substance, how much will remain after 30 years?

1. Type: Exponential Decay

2. Known Variables: We need to find the decay rate first. The half-life formula is:

   A = P(1/2)^(t/h)

   Where h is the half-life.

3. Finding the decay rate: After 10 years (one half-life), 50 grams will remain. Let's use the general exponential decay formula to find r:

   50 = 100(1 - r)^10

   0.5 = (1 - r)^10

   (0.5)^(1/10) = 1 - r

   r ≈ 0.067 (approximately 6.7% decay rate)

4. Calculating the amount after 30 years: Now use the general decay formula with r ≈ 0.067 and t = 30:

   A = 100(1 - 0.067)^30

5. Solution: A ≈ 12.5 grams

6. Interpretation: Approximately 12.5 grams of the substance will remain after 30 years. Alternatively, since 30 years is three half-lives, you can directly calculate this as 100g * (1/2)^3 = 12.5g.

Example 4: Depreciation

A car is purchased for $20,000 and depreciates at a rate of 15% per year. What will its value be after 5 years?

1. Type: Exponential Decay

2. Known Variables: P = $20,000, r = -0.15, t = 5

3. Formula: A = P(1 + r)^t

4. Substitution: A = 20000(1 - 0.15)^5

5. Solution: A = 20000(0.85)^5 ≈ $8875.0

6. Interpretation: The car will be worth approximately $8875 after 5 years.

Handling More Complex Scenarios: Solving for Different Variables

The examples above primarily focused on solving for the final amount (A). However, growth and decay problems can also require solving for the initial amount (P), the rate (r), or the time (t). Solving for these variables often involves the use of logarithms.

Solving for t: This involves taking the logarithm of both sides of the equation. For example, if you have A = P(1+r)^t and need to solve for t:

log(A/P) = t * log(1+r)

t = log(A/P) / log(1+r)

Solving for r: This requires a similar approach using logarithms, though the algebraic manipulation will be more complex.

Frequently Asked Questions (FAQ)

Q1: What is the difference between linear and exponential growth/decay?

A: Linear growth/decay involves a constant additive change over time (e.g., increasing by 5 units each year). Exponential growth/decay involves a constant multiplicative change over time (e.g., increasing by 5% each year). Exponential growth/decay results in much faster increases or decreases over time compared to linear growth/decay.

Q2: How do I handle problems with continuous compounding?

A: For continuous compounding (like in continuous population growth models or certain investment scenarios), a different formula is used:

A = Pe^(rt)

Where e is the mathematical constant approximately equal to 2.71828.

Q3: What if the growth or decay rate is not constant?

A: If the growth or decay rate changes over time, the simple exponential formulas are no longer applicable. More sophisticated mathematical models, such as differential equations, would be needed to accurately represent the situation.

Conclusion

Growth and decay word problems are a fundamental application of exponential functions with widespread practical applications. By understanding the underlying principles, the step-by-step solution process, and the use of logarithms when solving for different variables, you can confidently approach and solve a wide range of these problems. Remember to always carefully analyze the problem statement, identify the known variables, choose the correct formula, and interpret your solution within the context of the problem. With practice, you'll become proficient in mastering these essential mathematical concepts.