The Initial Population Size Of An Animal Species Is Measured To Be 2000. The Population Doubles Every 8 Years. Which Of The Following Functions Gives The Time, In Years, As An Output Value, And A Certain Number X X X For The Population Size As

Introduction

In this article, we will explore the concept of exponential growth and its application to a real-world scenario. The initial population size of an animal species is measured to be 2000, and it doubles every 8 years. We will analyze the situation and determine which function gives the time, in years, as an output value, and a certain number $x$ for the population size.

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 constant, and the value increases exponentially over time. The general formula for exponential growth is:

$$y = ab^x$$

where $y$ is the final value, $a$ is the initial value, $b$ is the growth factor, and $x$ is the time.

The Population Growth Model

In this scenario, the initial population size is 2000, and it doubles every 8 years. This means that the growth factor $b$ is 2, and the time period $x$ is 8 years. We can use the exponential growth formula to model the population growth:

$$P = 2000 \cdot 2^{t/8}$$

where $P$ is the population size, and $t$ is the time in years.

Determining the Time Function

We are asked to determine which function gives the time, in years, as an output value, and a certain number $x$ for the population size. In other words, we need to find the function that takes the population size $x$ as input and returns the corresponding time $t$.

To do this, we can rearrange the population growth formula to solve for $t$:

$$2^{t/8} = \frac{x}{2000}$$

Taking the logarithm base 2 of both sides, we get:

$$\frac{t}{8} = \log_2 \left(\frac{x}{2000}\right)$$

Multiplying both sides by 8, we get:

$$t = 8 \log_2 \left(\frac{x}{2000}\right)$$

This is the function that takes the population size $x$ as input and returns the corresponding time $t$.

Conclusion

In this article, we analyzed the population growth of an animal species and determined the function that gives the time, in years, as an output value, and a certain number $x$ for the population size. The function is given by:

$$t = 8 \log_2 \left(\frac{x}{2000}\right)$$

This function can be used to calculate the time it takes for the population to reach a certain size, given the initial population size and the growth rate.

Mathematical Derivations

Derivation of the Time Function

We start with the population growth formula:

$$P = 2000 \cdot 2^{t/8}$$

We can rearrange this formula to solve for $t$:

$$2^{t/8} = \frac{P}{2000}$$

Taking the logarithm base 2 of both sides, we get:

$$\frac{t}{8} = \log_2 \left(\frac{P}{2000}\right)$$

Multiplying both sides by 8, we get:

$$t = 8 \log_2 \left(\frac{P}{2000}\right)$$

This is the function that takes the population size $P$ as input and returns the corresponding time $t$.

Derivation of the Population Size Function

We can also derive the population size function by rearranging the population growth formula:

$$P = 2000 \cdot 2^{t/8}$$

Dividing both sides by 2000, we get:

$$\frac{P}{2000} = 2^{t/8}$$

Taking the logarithm base 2 of both sides, we get:

$$\log_2 \left(\frac{P}{2000}\right) = \frac{t}{8}$$

Multiplying both sides by 8, we get:

$$8 \log_2 \left(\frac{P}{2000}\right) = t$$

This is the function that takes the time $t$ as input and returns the corresponding population size $P$.

Applications of the Time Function

The time function can be used in a variety of applications, such as:

• Predicting population growth: The time function can be used to predict the time it takes for the population to reach a certain size, given the initial population size and the growth rate.
• Optimizing resource allocation: The time function can be used to optimize resource allocation by determining the optimal time to allocate resources to the population.
• Understanding population dynamics: The time function can be used to understand the dynamics of population growth and how it is affected by various factors such as growth rate, initial population size, and time.

Limitations of the Time Function

The time function has several limitations, including:

• Assumes exponential growth: The time function assumes that the population grows exponentially, which may not always be the case.
• Does not account for external factors: The time function does not account for external factors such as predation, disease, and environmental changes that can affect population growth.
• Requires accurate data: The time function requires accurate data on the initial population size, growth rate, and time period, which can be difficult to obtain.

Conclusion

Q: What is the initial population size of the animal species?

A: The initial population size of the animal species is 2000.

Q: How often does the population double?

A: The population doubles every 8 years.

Q: What is the growth factor of the population?

A: The growth factor of the population is 2.

Q: What is the formula for the population growth?

A: The formula for the population growth is:

$$P = 2000 \cdot 2^{t/8}$$

where $P$ is the population size, and $t$ is the time in years.

Q: How can we determine the time function that takes the population size $x$ as input and returns the corresponding time $t$?

A: We can determine the time function by rearranging the population growth formula:

$$2^{t/8} = \frac{x}{2000}$$

Taking the logarithm base 2 of both sides, we get:

$$\frac{t}{8} = \log_2 \left(\frac{x}{2000}\right)$$

Multiplying both sides by 8, we get:

$$t = 8 \log_2 \left(\frac{x}{2000}\right)$$

Q: What is the time function that takes the population size $x$ as input and returns the corresponding time $t$?

A: The time function is given by:

$$t = 8 \log_2 \left(\frac{x}{2000}\right)$$

Q: Can we use the time function to predict the time it takes for the population to reach a certain size?

A: Yes, we can use the time function to predict the time it takes for the population to reach a certain size, given the initial population size and the growth rate.

Q: What are some applications of the time function?

A: Some applications of the time function include:

• Predicting population growth: The time function can be used to predict the time it takes for the population to reach a certain size, given the initial population size and the growth rate.
• Optimizing resource allocation: The time function can be used to optimize resource allocation by determining the optimal time to allocate resources to the population.
• Understanding population dynamics: The time function can be used to understand the dynamics of population growth and how it is affected by various factors such as growth rate, initial population size, and time.

Q: What are some limitations of the time function?

A: Some limitations of the time function include:

• Assumes exponential growth: The time function assumes that the population grows exponentially, which may not always be the case.
• Does not account for external factors: The time function does not account for external factors such as predation, disease, and environmental changes that can affect population growth.
• Requires accurate data: The time function requires accurate data on the initial population size, growth rate, and time period, which can be difficult to obtain.

Q: Can we use the time function to understand the dynamics of population growth?

A: Yes, we can use the time function to understand the dynamics of population growth and how it is affected by various factors such as growth rate, initial population size, and time.

Q: What are some real-world applications of the time function?

A: Some real-world applications of the time function include:

• Conservation biology: The time function can be used to understand the dynamics of population growth and how it is affected by various factors such as habitat loss, predation, and disease.
• Epidemiology: The time function can be used to understand the dynamics of disease spread and how it is affected by various factors such as population size, growth rate, and time.
• Ecology: The time function can be used to understand the dynamics of ecosystem growth and how it is affected by various factors such as population size, growth rate, and time.