Notes on Exponential Growth and Decay Problems

Introduction

• Focus on solving exponential growth and decay word problems.
• Example problems involving rabbits, car depreciation, home value, and bacteria growth.

Exponential Growth

• Formula: ( y = a \times b^x )
  • a: Initial amount
  • b: Growth factor (1 + rate)
  • x: Time in relevant units

Problem 1: Rabbit Population

• Initial Population: 1000 rabbits in 2005
• Growth Rate: 8% per year
• Time Period: 2005 to 2020 (15 years)
• Equation: ( y = 1000 \times (1.08)^{15} )
• Solution: Approximately 3,172 rabbits by 2020

Exponential Decay

• Formula: ( y = a \times (1 - r)^t )
  • r: Decay rate

Problem 2: Car Depreciation

• Initial Value: $40,000 in 2015
• Depreciation Rate: 7% per year
• Time Period: 2015 to 2024 (9 years)
• Equation: ( y = 40000 \times (0.93)^9 )
• Solution: Approximately $2,816.44 by 2024

Reverse Calculation Using Exponential Growth

Problem 3: Home Value

• Present Value: $225,000 in 2015
• Growth Rate: 4% per year
• Time Period: 2002 to 2015 (13 years)
• Equation for Finding Initial Value: ( y = a \times (1.04)^{13} )
• Initial Value Solution: Approximately $135,129.17 in 2002

Bacteria Growth and Doubling Time

Problem 4: Bacteria Doubling

• Initial Count: 1000 bacteria
• Doubling Time: 20 minutes
• Time Period: 3 hours
• Converting Units:
  • 1 hour = 60 minutes
  • 3 hours = 180 minutes
  • Doubling 3 times an hour
• Equation: ( y = 1000 \times 2^{9} )
• Solution: 512,000 counts of bacteria

Bacteria Tripling and Solving for Time

Problem 5: Bacteria Tripling

• Initial Count: 100 bacteria
• Tripling Time: 15 minutes
• Conversion:
  • 1 hour = 60 minutes
  • Triples 4 times in an hour
• Equation: ( y = 100 \times 3^{4t} )
• Part A: Bacteria in 1 hour
  • Solution: 8,100 bacteria
• Part B: Time for 500 million bacteria
  • Using Logarithms
  • Solution: Approximately 3.51 hours

Conclusion

• Understanding exponential growth and decay formulas is crucial for solving related word problems.
• Practice with real-world examples such as population growth, depreciation, and microbial growth enhances comprehension.