The Quadrupling Population Model

Oct 13, 2025 by windailys.com 126 views

Introduction

Biologists are tracking the growth of a rabbit population, which is quadrupling every year. The initial population of rabbits is 45, and we need to find out when the population will reach 720. In this article, we will explore the mathematical equations that can be used to solve this problem.

The Quadrupling Population Model

The population of rabbits is quadrupling every year, which means that the population at the end of each year is four times the population at the end of the previous year. This can be represented by the equation:

P(t) = P0 * 4^t

where P(t) is the population at time t, P0 is the initial population, and 4^t is the growth factor.

Finding the Time When the Population Reaches 720

We want to find the time when the population of rabbits will be equal to 720. We can set up an equation using the quadrupling population model:

720 = 45 * 4^t

To solve for t, we can divide both sides of the equation by 45:

16 = 4^t

Now, we can take the logarithm base 4 of both sides of the equation to solve for t:

t = log4(16)

Using the change of base formula, we can rewrite the logarithm as:

t = log(16) / log(4)

Using a calculator, we can find that:

t ≈ 2.12

Alternative Equations

There are other equations that can be used to find the time when the population of rabbits will be equal to 720. One of these equations is:

P(t) = P0 * (4^t - 1) / 3

This equation is derived from the fact that the population at the end of each year is four times the population at the end of the previous year, minus the initial population.

We can set up an equation using this model:

720 = 45 * (4^t - 1) / 3

To solve for t, we can multiply both sides of the equation by 3:

2160 = 45 * (4^t - 1)

Now, we can divide both sides of the equation by 45:

48 = 4^t - 1

Adding 1 to both sides of the equation, we get:

49 = 4^t

Now, we can take the logarithm base 4 of both sides of the equation to solve for t:

t = log4(49)

Using the change of base formula, we can rewrite the logarithm as:

t = log(49) / log(4)

Using a calculator, we can find that:

t ≈ 2.12

Conclusion

In this article, we have explored the mathematical equations that can be used to find the time when the population of rabbits will be equal to 720. We have used the quadrupling population model and an alternative equation to solve for t. The results show that the population of rabbits will reach 720 in approximately 2.12 years.

References

[1] "Quadrupling Population Model". Wikipedia.
[2] "Logarithm". Wikipedia.

Mathematical Formulas

P(t) = P0 * 4^t
P(t) = P0 * (4^t - 1) / 3
t = log4(16)
t = log(16) / log(4)
t = log4(49)
t = log(49) / log(4)
Biologists are tracking the growth of a rabbit population: Q&A =============================================================

Introduction

In our previous article, we explored the mathematical equations that can be used to find the time when the population of rabbits will be equal to 720. 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 45.

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

A: The rabbit population is quadrupling every year, which means that the population at the end of each year is four times the population at the end of the previous year.

Q: How can we represent the population growth of the rabbits mathematically?

A: We can use the equation P(t) = P0 * 4^t, where P(t) is the population at time t, P0 is the initial population, and 4^t is the growth factor.

Q: How can we find the time when the population of rabbits will be equal to 720?

A: We can set up an equation using the quadrupling population model: 720 = 45 * 4^t. To solve for t, we can divide both sides of the equation by 45, and then take the logarithm base 4 of both sides of the equation.

Q: What is the time when the population of rabbits will be equal to 720?

A: Using the equation P(t) = P0 * 4^t, we can find that t ≈ 2.12 years.

Q: Are there other equations that can be used to find the time when the population of rabbits will be equal to 720?

A: Yes, there is an alternative equation that can be used: P(t) = P0 * (4^t - 1) / 3. We can set up an equation using this model and solve for t.

Q: How can we use the alternative equation to find the time when the population of rabbits will be equal to 720?

A: We can set up an equation using the alternative model: 720 = 45 * (4^t - 1) / 3. To solve for t, we can multiply both sides of the equation by 3, and then divide both sides of the equation by 45. We can then add 1 to both sides of the equation, and take the logarithm base 4 of both sides of the equation.

Q: What is the time when the population of rabbits will be equal to 720 using the alternative equation?

A: Using the alternative equation, we can find that t ≈ 2.12 years.

Q: Why is the time when the population of rabbits will be equal to 720 the same using both equations?

A: The time when the population of rabbits will be equal to 720 is the same using both equations because both equations are based on the same population growth model. The quadrupling population model and the alternative equation are equivalent, and they will give the same result.

Conclusion

In this article, we have answered some frequently asked questions about the rabbit population growth problem. We have used the quadrupling population model and an alternative equation to find the time when the population of rabbits will be equal to 720. The results show that the population of rabbits will reach 720 in approximately 2.12 years.

References

[1] "Quadrupling Population Model". Wikipedia.
[2] "Logarithm". Wikipedia.

Mathematical Formulas

P(t) = P0 * 4^t
P(t) = P0 * (4^t - 1) / 3
t = log4(16)
t = log(16) / log(4)
t = log4(49)
t = log(49) / log(4)