Type The Correct Answer In Each Box. Use Numerals Instead Of Words.The Population Of Rabbits In A Park Is Modeled By R ( X ) = 34 ( 1.85 ) X R(x) = 34(1.85)^x R ( X ) = 34 ( 1.85 ) X , Where X X X Represents The Number Of Years Since The Counting Began.The Initial Number Of Rabbits

Mar 10, 2025 by ADMIN 283 views

**Population Modeling of Rabbits in a Park**

Introduction

The population of rabbits in a park is modeled by the exponential function $r(x) = 34(1.85)^x$ , where $x$ represents the number of years since the counting began. In this article, we will explore the population growth of rabbits in the park and determine the initial number of rabbits.

Understanding the Exponential Function

The exponential function $r(x) = 34(1.85)^x$ represents the population of rabbits in the park at any given time $x$ . The base of the exponential function is 1.85, which represents the growth rate of the rabbit population. The initial population of rabbits is given by the value of the function when $x = 0$ , which is $r(0) = 34(1.85)^0 = 34$ .

Determining the Initial Number of Rabbits

To determine the initial number of rabbits, we need to find the value of the function when $x = 0$ . This can be done by substituting $x = 0$ into the function:

$r(0) = 34(1.85)^0$

Since any number raised to the power of 0 is equal to 1, we have:

$r(0) = 34(1)^0 = 34$

Therefore, the initial number of rabbits in the park is 34.

Calculating the Population at a Given Time

To calculate the population of rabbits at a given time $x$ , we can substitute the value of $x$ into the function:

$r(x) = 34(1.85)^x$

For example, to calculate the population of rabbits after 5 years, we can substitute $x = 5$ into the function:

$r(5) = 34(1.85)^5$

Using a calculator, we can evaluate the expression:

$r(5) = 34(1.85)^5 ≈ 34(5.59) ≈ 190.06$

Therefore, the population of rabbits in the park after 5 years is approximately 190.

Calculating the Population at a Given Time (continued)

To calculate the population of rabbits at a given time $x$ , we can also use the formula:

$r(x) = 34(1.85)^x$

For example, to calculate the population of rabbits after 10 years, we can substitute $x = 10$ into the function:

$r(10) = 34(1.85)^10$

Using a calculator, we can evaluate the expression:

$r(10) = 34(1.85)^10 ≈ 34(14.59) ≈ 496.06$

Therefore, the population of rabbits in the park after 10 years is approximately 496.

Conclusion

In this article, we have explored the population growth of rabbits in a park modeled by the exponential function $r(x) = 34(1.85)^x$ . We have determined the initial number of rabbits to be 34 and calculated the population at a given time using the formula. The population of rabbits in the park is growing exponentially, and the number of rabbits will continue to increase over time.

Population Growth of Rabbits in the Park

Year	Population
0	34
5	190.06
10	496.06
15	1,144.06
20	2,624.06

Table 1: Population of rabbits in the park at different times.

Population Growth Rate of Rabbits in the Park

The population growth rate of rabbits in the park is given by the base of the exponential function, which is 1.85. This means that the population of rabbits is growing at a rate of 85% per year.

Conclusion

In conclusion, the population of rabbits in a park is modeled by the exponential function $r(x) = 34(1.85)^x$ . The initial number of rabbits is 34, and the population is growing exponentially at a rate of 85% per year. The population of rabbits in the park will continue to increase over time, and the number of rabbits will reach 1,000 in approximately 15 years.

References

[1] "Exponential Functions." Math Open Reference, mathopenref.com/exponential.html.
[2] "Population Growth." Khan Academy, khanacademy.org/math/ap-calculus-ab/population-growth.

Glossary

Exponential function: A function of the form $f(x) = ab^x$ , where $a$ and $b$ are constants.
Population growth rate: The rate at which the population of a species is increasing or decreasing.
Base: The constant $b$ in the exponential function $f(x) = ab^x$ .
Exponent: The constant $x$ in the exponential function $f(x) = ab^x$ .
Q&A: Population Modeling of Rabbits in a Park =====================================================

Frequently Asked Questions

Q: What is the initial number of rabbits in the park?

A: The initial number of rabbits in the park is 34.

Q: How is the population of rabbits in the park modeled?

A: The population of rabbits in the park is modeled by the exponential function $r(x) = 34(1.85)^x$ , where $x$ represents the number of years since the counting began.

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

A: The growth rate of the rabbit population is 85% per year, which is given by the base of the exponential function, 1.85.

Q: How can I calculate the population of rabbits at a given time?

A: To calculate the population of rabbits at a given time $x$ , you can substitute the value of $x$ into the function $r(x) = 34(1.85)^x$ .

Q: What is the population of rabbits in the park after 5 years?

A: The population of rabbits in the park after 5 years is approximately 190.06.

Q: What is the population of rabbits in the park after 10 years?

A: The population of rabbits in the park after 10 years is approximately 496.06.

Q: How long will it take for the population of rabbits in the park to reach 1,000?

A: It will take approximately 15 years for the population of rabbits in the park to reach 1,000.

Q: What is the formula for calculating the population of rabbits at a given time?

A: The formula for calculating the population of rabbits at a given time $x$ is $r(x) = 34(1.85)^x$ .

Q: What is the base of the exponential function that models the population of rabbits in the park?

A: The base of the exponential function that models the population of rabbits in the park is 1.85.

Q: What is the exponent in the exponential function that models the population of rabbits in the park?

A: The exponent in the exponential function that models the population of rabbits in the park is $x$ , where $x$ represents the number of years since the counting began.

Q: What is the constant term in the exponential function that models the population of rabbits in the park?

A: The constant term in the exponential function that models the population of rabbits in the park is 34.

Q: What is the significance of the population growth rate in the context of the rabbit population model?

A: The population growth rate is a measure of how quickly the rabbit population is increasing or decreasing. In this case, the population growth rate is 85% per year, which means that the rabbit population is growing rapidly.

Q: How can I use the rabbit population model to make predictions about the future population of rabbits in the park?

A: You can use the rabbit population model to make predictions about the future population of rabbits in the park by substituting different values of $x$ into the function $r(x) = 34(1.85)^x$ and calculating the resulting population.

Q: What are some potential limitations of the rabbit population model?

A: Some potential limitations of the rabbit population model include the assumption that the population growth rate remains constant over time, the assumption that the population is not affected by external factors such as disease or predation, and the assumption that the population is not limited by resources such as food or space.

Q: How can I modify the rabbit population model to account for these limitations?

A: You can modify the rabbit population model to account for these limitations by incorporating additional variables or parameters into the model, such as a variable population growth rate or a limit on the population size.