A Second Sample Of 20,000 Bacteria Is Being Observed. The Equation Used To Represent The Population Of Sample 2 After { T$}$ Days Is { \rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$.Which Of The Following Is An Equivalent Form Of The

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Introduction

In the field of mathematics, particularly in the realm of exponential growth, understanding the behavior of populations is crucial. The equation used to represent the population of sample 2 after {t$}$ days is {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$. This equation is a representation of the population growth of a sample of 20,000 bacteria. In this article, we will delve into the world of exponential growth and explore the equivalent forms of the given equation.

Understanding Exponential Growth

Exponential growth is a type of growth where the rate of growth is proportional to the current value. In the context of the bacteria population, the equation {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$ represents the population of the bacteria after {t$}$ days. The base of the exponent, 2, represents the growth factor, which is the rate at which the population is increasing.

The Given Equation

The equation {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$ can be rewritten in a more simplified form. To do this, we need to understand the properties of exponents.

Properties of Exponents

Exponents have several properties that can be used to simplify expressions. One of the most important properties is the power rule, which states that {a{m}a{n} = a^{m+n}$}$. Another property is the product rule, which states that {a{m}b{m} = (ab)^{m}$}$.

Simplifying the Equation

Using the properties of exponents, we can simplify the equation {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$. We can rewrite the equation as {\rho(t) = 20,000(2){\frac{t}{10}}(2){-\frac{7}{10}}$}$.

Applying the Product Rule

Using the product rule, we can rewrite the equation as {\rho(t) = (20,000)(2){\frac{t}{10}}(2){-\frac{7}{10}}$}$.

Applying the Power Rule

Using the power rule, we can rewrite the equation as {\rho(t) = (20,000)(2)^{\frac{t-7}{10}}$}$.

Equivalent Forms of the Equation

The equation {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$ has several equivalent forms. One of the equivalent forms is {\rho(t) = (20,000)(2)^{\frac{t-7}{10}}$}$.

Conclusion

In conclusion, the equation {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$ represents the population growth of a sample of 20,000 bacteria. The equation has several equivalent forms, including {\rho(t) = (20,000)(2)^{\frac{t-7}{10}}$}$. Understanding the properties of exponents and applying them to simplify expressions is crucial in mathematics, particularly in the realm of exponential growth.

References

Further Reading

Glossary

  • Exponential growth: A type of growth where the rate of growth is proportional to the current value.
  • Exponent: A number or expression that is raised to a power.
  • Power rule: A property of exponents that states {a{m}a{n} = a^{m+n}$}$.
  • Product rule: A property of exponents that states {a{m}b{m} = (ab)^{m}$}$.
    A Second Sample of Bacteria: Understanding the Population Growth Equation - Q&A ====================================================================================

Introduction

In our previous article, we explored the equation {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$ and its equivalent forms. This equation represents the population growth of a sample of 20,000 bacteria. In this article, we will answer some frequently asked questions related to the equation and its applications.

Q&A

Q: What is the initial population of the bacteria?

A: The initial population of the bacteria is 20,000.

Q: What is the growth factor of the bacteria?

A: The growth factor of the bacteria is 2, which means that the population of the bacteria doubles every 10 days.

Q: What is the equation used to represent the population of the bacteria after t days?

A: The equation used to represent the population of the bacteria after t days is {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$.

Q: What is the equivalent form of the equation?

A: The equivalent form of the equation is {\rho(t) = (20,000)(2)^{\frac{t-7}{10}}$}$.

Q: How can we use the equation to predict the population of the bacteria after a certain number of days?

A: We can use the equation to predict the population of the bacteria after a certain number of days by plugging in the value of t into the equation.

Q: What is the significance of the exponent in the equation?

A: The exponent in the equation represents the growth factor of the bacteria. It indicates how quickly the population of the bacteria is increasing.

Q: Can we use the equation to model the population growth of other organisms?

A: Yes, we can use the equation to model the population growth of other organisms, such as animals or plants, as long as the growth factor is known.

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

A: Some real-world applications of the equation include modeling the population growth of bacteria in a laboratory setting, predicting the population growth of animals in the wild, and understanding the spread of diseases.

Conclusion

In conclusion, the equation {\rho(t) = 20,000(2)^{\frac{t-7}{10}}$}$ represents the population growth of a sample of 20,000 bacteria. Understanding the equation and its equivalent forms can help us predict the population of the bacteria after a certain number of days and model the population growth of other organisms.

References

Further Reading

Glossary

  • Exponential growth: A type of growth where the rate of growth is proportional to the current value.
  • Exponent: A number or expression that is raised to a power.
  • Power rule: A property of exponents that states {a{m}a{n} = a^{m+n}$}$.
  • Product rule: A property of exponents that states {a{m}b{m} = (ab)^{m}$}$.