The Number Of Bacteria Grown In A Lab Can Be Modeled By $P(t)=300 \cdot 2^{4t}$, Where $t$ Is The Number Of Hours. Which Expression Is Equivalent To $P(t$\]?1) $300 \cdot 8^t$ 2) $300 \cdot 16^t$ 3)

Feb 28, 2025 by ADMIN 201 views

**The Number of Bacteria Grown in a Lab: Modeling with Exponential Functions**

Introduction

In the field of mathematics, exponential functions are used to model various real-world phenomena, including population growth, chemical reactions, and bacterial growth. In this article, we will explore the exponential function $P(t)=300 \cdot 2^{4t}$ , which models the number of bacteria grown in a lab over time. We will examine the given expression and determine which alternative expression is equivalent to it.

Understanding the Exponential Function

The given exponential function is $P(t)=300 \cdot 2^{4t}$ . This function represents the number of bacteria grown in a lab at time $t$ , where $t$ is measured in hours. The base of the exponential function is 2, and the coefficient is 300. The exponent is $4t$ , indicating that the number of bacteria grows exponentially with time.

Analyzing the Alternative Expressions

We are given three alternative expressions to compare with the original expression:

$300 \cdot 8^t$
$300 \cdot 16^t$
(no alternative expression provided)

To determine which expression is equivalent to the original expression, we need to analyze each alternative expression separately.

Alternative Expression 1: $300 \cdot 8^t$

The first alternative expression is $300 \cdot 8^t$ . To compare this expression with the original expression, we need to rewrite the base 8 as a power of 2. Since $8=2^3$ , we can rewrite the expression as:

300 \cdot 8^t = 300 \cdot (2^3)^t = 300 \cdot 2^{3t}

Comparing this expression with the original expression, we can see that the exponent is different. The original expression has an exponent of $4t$ , while the rewritten expression has an exponent of $3t$ . Therefore, the first alternative expression is not equivalent to the original expression.

Alternative Expression 2: $300 \cdot 16^t$

The second alternative expression is $300 \cdot 16^t$ . To compare this expression with the original expression, we need to rewrite the base 16 as a power of 2. Since $16=2^4$ , we can rewrite the expression as:

300 \cdot 16^t = 300 \cdot (2^4)^t = 300 \cdot 2^{4t}

Comparing this expression with the original expression, we can see that the two expressions are identical. Therefore, the second alternative expression is equivalent to the original expression.

Conclusion

In conclusion, the expression equivalent to $P(t)=300 \cdot 2^{4t}$ is $300 \cdot 16^t$ . This expression models the number of bacteria grown in a lab over time, with the base 16 representing the exponential growth of the bacteria population.

Discussion

The exponential function $P(t)=300 \cdot 2^{4t}$ is a common model used in biology to describe the growth of bacterial populations. The equivalent expression $300 \cdot 16^t$ provides a simpler way to represent the same growth pattern. This equivalence highlights the importance of understanding the properties of exponential functions and their applications in real-world scenarios.

References

[1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
[2] "Bacterial Growth" by Khan Academy. Retrieved from https://www.khanacademy.org/science/biology/growth-and-reproduction-of-bacteria/e/bacterial-growth

Keywords

Exponential functions
Bacterial growth
Population modeling
Mathematical modeling
Equivalent expressions
The Number of Bacteria Grown in a Lab: Modeling with Exponential Functions ================================================================================

Q&A: Understanding Exponential Functions and Bacterial Growth

Introduction

In our previous article, we explored the exponential function $P(t)=300 \cdot 2^{4t}$ , which models the number of bacteria grown in a lab over time. We also determined that the expression equivalent to this function is $300 \cdot 16^t$ . In this article, we will answer some frequently asked questions about exponential functions and bacterial growth.

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form $f(x) = ab^x$ , where $a$ and $b$ are constants, and $x$ is the variable. Exponential functions are used to model various real-world phenomena, including population growth, chemical reactions, and bacterial growth.

Q: How do exponential functions model bacterial growth?

A: Exponential functions model bacterial growth by representing the number of bacteria as a function of time. The base of the exponential function represents the growth rate of the bacteria, while the exponent represents the time at which the bacteria are measured.

Q: What is the significance of the base 2 in the exponential function $P(t)=300 \cdot 2^{4t}$ ?

A: The base 2 in the exponential function $P(t)=300 \cdot 2^{4t}$ represents the growth rate of the bacteria. In this case, the bacteria are growing at a rate of 2 times the previous population every hour.

Q: How does the exponent $4t$ affect the growth of the bacteria?

A: The exponent $4t$ represents the time at which the bacteria are measured. As time increases, the exponent also increases, resulting in a rapid growth of the bacteria population.

Q: What is the equivalent expression to $P(t)=300 \cdot 2^{4t}$ ?

A: The equivalent expression to $P(t)=300 \cdot 2^{4t}$ is $300 \cdot 16^t$ . This expression represents the same growth pattern as the original expression, but with a simpler base.

Q: How can exponential functions be used in real-world applications?

A: Exponential functions can be used in various real-world applications, including population modeling, chemical reactions, and financial modeling. In the context of bacterial growth, exponential functions can be used to model the growth of bacteria in a lab or in a natural environment.

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

Confusing the base and the exponent
Failing to simplify the expression
Not considering the domain and range of the function

Conclusion

In conclusion, exponential functions are a powerful tool for modeling various real-world phenomena, including bacterial growth. By understanding the properties of exponential functions and their applications, we can better model and analyze complex systems.

Discussion

References

[1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
[2] "Bacterial Growth" by Khan Academy. Retrieved from https://www.khanacademy.org/science/biology/growth-and-reproduction-of-bacteria/e/bacterial-growth

Keywords

Exponential functions
Bacterial growth
Population modeling
Mathematical modeling
Equivalent expressions
Real-world applications

Introduction

Understanding the Exponential Function

Analyzing the Alternative Expressions

Alternative Expression 1: 300⋅8t300 \cdot 8^t300⋅8t

Alternative Expression 2: 300⋅16t300 \cdot 16^t300⋅16t

Conclusion

Discussion

References

Keywords

Q&A: Understanding Exponential Functions and Bacterial Growth

Introduction

Q: What is an exponential function?

Q: How do exponential functions model bacterial growth?

Q: What is the significance of the base 2 in the exponential function P(t)=300⋅24tP(t)=300 \cdot 2^{4t}P(t)=300⋅24t?

Q: How does the exponent 4t4t4t affect the growth of the bacteria?

Q: What is the equivalent expression to P(t)=300⋅24tP(t)=300 \cdot 2^{4t}P(t)=300⋅24t?

Q: How can exponential functions be used in real-world applications?

Q: What are some common mistakes to avoid when working with exponential functions?

Conclusion

Discussion

References

Keywords

Alternative Expression 1: $300 \cdot 8^t$

Alternative Expression 2: $300 \cdot 16^t$

Q: What is the significance of the base 2 in the exponential function $P(t)=300 \cdot 2^{4t}$ ?

Q: How does the exponent $4t$ affect the growth of the bacteria?

Q: What is the equivalent expression to $P(t)=300 \cdot 2^{4t}$ ?