How to Solve Exponential Growth and Decay Problems

TL;DR

To solve exponential growth problems, use the formula y = a(1 + r)^t, where 'a' is the initial value, 'r' is the growth rate, and 't' is time. For decay, the formula is y = a(1 - r)^t. Understanding how to apply these formulas allows you to calculate future populations or values effectively.

Transcript

in this video we're going to focus on solving exponential growth and decay word problems so let's get right into it in 2005 there were a thousand rabbits on an island the population grows eight percent every year at this rate how many rabbits will there be on the island by 2020 so what should we do in this problem how can we find the answer so we n... Read More

Key Insights

• ⌛ Exponential growth problems involve an increase in value or population over time, while exponential decay problems involve a decrease.
• 🤘 The formulas for exponential growth and decay are similar, but the sign in the exponent differs.
• ⌛ Calculating the number of times a value doubles or triples can be done by dividing the total time by the time it takes for one doubling or tripling to occur.
• ❓ The initial value or population is represented by the variable 'a' in the formulas.
• 🔌 Converting percentages to decimals is necessary when plugging values into the formulas.
• 🤩 Key information needed for solving exponential growth and decay problems include initial value, growth/decay rate, and time period.
• ⌛ Logarithms can be used to solve for the time 't' in exponential growth or decay problems.

Questions & Answers

Q: What is the formula for exponential growth and decay problems?

The formula for exponential growth is y = a times b raised to the x, while the formula for exponential decay is y = a times b raised to the -x. These formulas vary based on whether the value is increasing or decreasing over time.

Q: How do you convert a percentage to a decimal?

To convert a percentage to a decimal, divide the percentage by 100. For example, 8% would be converted to 0.08 by dividing 8 by 100.

Q: What does the variable 'a' represent in exponential growth and decay problems?

In exponential growth problems, 'a' represents the initial value or population. In the examples, 'a' represented the initial population of rabbits, the initial value of the car, and the initial price of the house.

Q: How do you calculate the number of years between two dates?

To calculate the number of years between two dates, subtract the earlier year from the later year. For example, if the earlier year is 2002 and the later year is 2015, the difference is 13 years.

Q: How do you determine the number of times a value doubles or triples in a certain time period?

Divide the total time period by the time it takes for the value to double or triple once. For example, if the value triples every 15 minutes and you want to calculate the number of triples in an hour, divide 60 minutes by 15 minutes, resulting in 4 triples.

Summary & Key Takeaways

• The first example deals with exponential growth, where the population of rabbits on an island increases by 8% every year. By 2020, there will be approximately 3,172 rabbits.

• The second example involves exponential decay, where the value of a car depreciates by 7% each year. In 2024, the car will be worth approximately $20,816.44.

• The third example focuses on exponential growth, where the value of a home increases by 4% each year. In 2002, the house was purchased for $135,129.17.