The Size \[$ P \$\] Of A Certain Insect Population At Time \[$ T \$\] (in Days) Obeys The Function \[$ P(t) = 900 E^{0.08 T} \$\].(a) Determine The Number Of Insects At \[$ T = 0 \$\] Days.(b) What Is The Growth Rate Of

Introduction

The study of population growth is a fundamental concept in mathematics, with applications in various fields such as biology, economics, and sociology. In this article, we will analyze the size of a certain insect population at time $t$ (in days) using the function $P(t) = 900 e^{0.08 t}$.

Understanding the Function

The given function $P(t) = 900 e^{0.08 t}$ represents the size of the insect population at time $t$. Here, $P(t)$ is the population size at time $t$, and $e^{0.08 t}$ is the exponential growth factor. The constant $900$ represents the initial population size, and the coefficient $0.08$ represents the growth rate of the population.

Part (a): Determining the Initial Population Size

To determine the initial population size at $t = 0$ days, we need to substitute $t = 0$ into the function $P(t) = 900 e^{0.08 t}$.

P(0) = 900 e^{0.08(0)}
P(0) = 900 e^0
P(0) = 900(1)
P(0) = 900

Therefore, the initial population size at $t = 0$ days is $900$ insects.

Part (b): Determining the Growth Rate

To determine the growth rate of the population, we need to analyze the coefficient $0.08$ in the function $P(t) = 900 e^{0.08 t}$. The growth rate is represented by the coefficient $0.08$, which is a decimal value between $0$ and $1$. In this case, the growth rate is $8\%$ per day.

Interpretation of the Growth Rate

The growth rate of $8\%$ per day means that the population size increases by $8\%$ every day. This is a relatively high growth rate, indicating that the population is growing rapidly. To put this into perspective, if the initial population size is $900$ insects, the population size will increase to $970$ insects after one day, $1047.2$ insects after two days, and so on.

Conclusion

In conclusion, the size of a certain insect population at time $t$ (in days) obeys the function $P(t) = 900 e^{0.08 t}$. By analyzing this function, we have determined the initial population size at $t = 0$ days to be $900$ insects and the growth rate of the population to be $8\%$ per day. This analysis provides valuable insights into the population growth dynamics of the insect population.

Mathematical Analysis of the Function

To further analyze the function $P(t) = 900 e^{0.08 t}$, we can use various mathematical techniques such as differentiation and integration.

Differentiation of the Function

To find the derivative of the function $P(t) = 900 e^{0.08 t}$, we can use the chain rule of differentiation.

\frac{dP}{dt} = 900 \cdot 0.08 e^{0.08 t}
\frac{dP}{dt} = 72 e^{0.08 t}

The derivative of the function represents the rate of change of the population size with respect to time.

Integration of the Function

To find the integral of the function $P(t) = 900 e^{0.08 t}$, we can use the formula for the integral of an exponential function.

\int P(t) dt = \int 900 e^{0.08 t} dt
\int P(t) dt = \frac{900}{0.08} e^{0.08 t} + C
\int P(t) dt = 11250 e^{0.08 t} + C

The integral of the function represents the cumulative population size over a given time period.

Applications of the Function

The function $P(t) = 900 e^{0.08 t}$ has various applications in real-world scenarios, such as:

• Population modeling: The function can be used to model the growth of a population over time, taking into account factors such as birth rates, death rates, and migration.
• Epidemiology: The function can be used to model the spread of diseases over time, taking into account factors such as transmission rates and recovery rates.
• Economics: The function can be used to model the growth of economies over time, taking into account factors such as interest rates and inflation rates.

Conclusion

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Introduction

In our previous article, we analyzed the size of a certain insect population at time $t$ (in days) using the function $P(t) = 900 e^{0.08 t}$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the initial population size of the insect population?

A: The initial population size of the insect population is $900$ insects, which is the value of the function $P(t)$ at $t = 0$ days.

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

A: The growth rate of the insect population is $8\%$ per day, which is represented by the coefficient $0.08$ in the function $P(t) = 900 e^{0.08 t}$.

Q: How does the population size change over time?

A: The population size increases exponentially over time, with a growth rate of $8\%$ per day. This means that the population size will double approximately every $9$ days.

Q: What is the population size after $t$ days?

A: The population size after $t$ days can be calculated using the function $P(t) = 900 e^{0.08 t}$.

Q: How can we use this function in real-world scenarios?

A: This function can be used to model the growth of a population over time, taking into account factors such as birth rates, death rates, and migration. It can also be used to model the spread of diseases over time, taking into account factors such as transmission rates and recovery rates.

Q: What are some limitations of this function?

A: One limitation of this function is that it assumes a constant growth rate, which may not be realistic in all scenarios. Additionally, the function does not take into account factors such as environmental changes, predation, and competition for resources.

Q: How can we modify this function to make it more realistic?

A: To make this function more realistic, we can add additional terms to account for factors such as environmental changes, predation, and competition for resources. We can also use more complex models, such as logistic growth models, to better capture the dynamics of population growth.

Q: What are some applications of this function in economics?

A: This function can be used to model the growth of economies over time, taking into account factors such as interest rates and inflation rates. It can also be used to model the spread of economic shocks, such as recessions and depressions.

Conclusion

In conclusion, the function $P(t) = 900 e^{0.08 t}$ represents the size of a certain insect population at time $t$ (in days). By answering some frequently asked questions related to this topic, we have provided a better understanding of the population growth dynamics of the insect population.

Frequently Asked Questions

• Q: What is the initial population size of the insect population? A: The initial population size of the insect population is $900$ insects.
• Q: What is the growth rate of the insect population? A: The growth rate of the insect population is $8\%$ per day.
• Q: How does the population size change over time? A: The population size increases exponentially over time, with a growth rate of $8\%$ per day.
• Q: What is the population size after $t$ days? A: The population size after $t$ days can be calculated using the function $P(t) = 900 e^{0.08 t}$.
• Q: How can we use this function in real-world scenarios? A: This function can be used to model the growth of a population over time, taking into account factors such as birth rates, death rates, and migration.
• Q: What are some limitations of this function? A: One limitation of this function is that it assumes a constant growth rate, which may not be realistic in all scenarios.
• Q: How can we modify this function to make it more realistic? A: To make this function more realistic, we can add additional terms to account for factors such as environmental changes, predation, and competition for resources.
• Q: What are some applications of this function in economics? A: This function can be used to model the growth of economies over time, taking into account factors such as interest rates and inflation rates.

Conclusion

In conclusion, the function $P(t) = 900 e^{0.08 t}$ represents the size of a certain insect population at time $t$ (in days). By answering some frequently asked questions related to this topic, we have provided a better understanding of the population growth dynamics of the insect population.