In The 1st Generation, There Are 3 Rabbits In A Meadow. Every Generation After That, The Rabbit Population Multiplies By 6. In Generation 2, There Are 18 Rabbits; In Generation 3, There Are 108 Rabbits, And So On.Which Explicit Formula Can Be Used To

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The problem of the rabbit population growth is a classic example of exponential growth in mathematics. In this article, we will explore the concept of exponential growth and derive an explicit formula to calculate the rabbit population in any given generation.

What is Exponential Growth?

Exponential growth is a type of growth where the rate of growth is proportional to the current value. In other words, the growth rate is not constant, but rather it increases as the value increases. This type of growth is often seen in real-world situations, such as population growth, financial growth, and chemical reactions.

The Rabbit Population Growth Problem

In the first generation, there are 3 rabbits in a meadow. Every generation after that, the rabbit population multiplies by 6. In generation 2, there are 18 rabbits; in generation 3, there are 108 rabbits, and so on. We need to find an explicit formula to calculate the rabbit population in any given generation.

Deriving the Explicit Formula

To derive the explicit formula, we can start by analyzing the pattern of the rabbit population growth. In the first generation, there are 3 rabbits. In the second generation, the population multiplies by 6, resulting in 18 rabbits. In the third generation, the population multiplies by 6 again, resulting in 108 rabbits.

We can see that the population is growing exponentially, with a growth rate of 6. To derive the explicit formula, we can use the formula for exponential growth:

A(t) = A0 * (1 + r)^t

where A(t) is the population at time t, A0 is the initial population, r is the growth rate, and t is the time.

In this case, the initial population A0 is 3, the growth rate r is 6, and the time t is the generation number. We can plug in these values into the formula to get:

A(t) = 3 * (1 + 6)^t A(t) = 3 * 7^t

This is the explicit formula for the rabbit population growth. We can use this formula to calculate the population in any given generation.

Example Calculations

Let's use the explicit formula to calculate the population in some example generations.

  • Generation 1: t = 1, A(1) = 3 * 7^1 = 21
  • Generation 2: t = 2, A(2) = 3 * 7^2 = 147
  • Generation 3: t = 3, A(3) = 3 * 7^3 = 1029
  • Generation 4: t = 4, A(4) = 3 * 7^4 = 7203

As we can see, the population is growing exponentially, with a growth rate of 6.

Conclusion

In this article, we have derived an explicit formula to calculate the rabbit population growth. The formula is A(t) = 3 * 7^t, where A(t) is the population at time t, and t is the generation number. We have also used the formula to calculate the population in some example generations. The rabbit population growth problem is a classic example of exponential growth, and the explicit formula can be used to model and analyze this type of growth in real-world situations.

References

  • [1] "Exponential Growth" by Khan Academy
  • [2] "Rabbit Population Growth" by Math Is Fun

Further Reading

If you want to learn more about exponential growth and the rabbit population growth problem, I recommend checking out the following resources:

  • Khan Academy's video on exponential growth
  • Math Is Fun's article on the rabbit population growth problem
  • The book "Exponential Growth and Decay" by Michael Corral

In the previous article, we derived an explicit formula to calculate the rabbit population growth. In this article, we will answer some frequently asked questions about the rabbit population growth problem.

Q: What is the initial population of rabbits?

A: The initial population of rabbits is 3.

Q: What is the growth rate of the rabbit population?

A: The growth rate of the rabbit population is 6.

Q: How do I calculate the population in a given generation?

A: To calculate the population in a given generation, you can use the explicit formula A(t) = 3 * 7^t, where A(t) is the population at time t, and t is the generation number.

Q: What is the population in generation 5?

A: To calculate the population in generation 5, we can plug in t = 5 into the explicit formula:

A(5) = 3 * 7^5 A(5) = 3 * 16807 A(5) = 50421

So, the population in generation 5 is 50421.

Q: How does the population grow exponentially?

A: The population grows exponentially because the growth rate is proportional to the current value. In other words, the growth rate is not constant, but rather it increases as the value increases.

Q: Can I use this formula to model other types of growth?

A: Yes, you can use this formula to model other types of growth that exhibit exponential behavior. However, you will need to adjust the initial population and growth rate to fit the specific problem you are trying to model.

Q: What are some real-world applications of exponential growth?

A: Exponential growth has many real-world applications, including:

  • Population growth: The growth of a population can be modeled using exponential growth.
  • Financial growth: The growth of investments or savings can be modeled using exponential growth.
  • Chemical reactions: The rate of a chemical reaction can be modeled using exponential growth.
  • Biological growth: The growth of living organisms can be modeled using exponential growth.

Q: How can I use this formula to solve problems in real-world situations?

A: To use this formula to solve problems in real-world situations, you will need to:

  • Identify the initial population and growth rate.
  • Plug in the values into the explicit formula.
  • Calculate the population at the desired time.

For example, if you are trying to model the growth of a population, you can use the formula to calculate the population at a given time. If you are trying to model the growth of an investment, you can use the formula to calculate the value of the investment at a given time.

Conclusion

In this article, we have answered some frequently asked questions about the rabbit population growth problem. We have also discussed some real-world applications of exponential growth and provided tips on how to use the formula to solve problems in real-world situations.

References

  • [1] "Exponential Growth" by Khan Academy
  • [2] "Rabbit Population Growth" by Math Is Fun

Further Reading

If you want to learn more about exponential growth and the rabbit population growth problem, I recommend checking out the following resources:

  • Khan Academy's video on exponential growth
  • Math Is Fun's article on the rabbit population growth problem
  • The book "Exponential Growth and Decay" by Michael Corral

I hope this article has been helpful in answering your questions about the rabbit population growth problem.