A Colony Contains 1500 Bacteria. The Population Increases At A Rate Of $115 \%$ Each Hour. If $x$ Represents The Number Of Hours Elapsed, Which Function Represents The Scenario?A. $f(x)=1500(1.15)^x$ B.

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Introduction

The growth of a bacterial colony is a classic example of exponential growth, where the population increases at a constant rate over time. In this scenario, we are given a colony of 1500 bacteria that increases at a rate of 115% each hour. We need to determine which function represents this scenario. To do this, we will analyze the characteristics of exponential growth and examine the given functions to see which one matches the description.

Exponential Growth

Exponential growth is a type of growth where the rate of increase is proportional to the current value. In other words, the more you have, the faster it grows. This type of growth is often modeled using exponential functions, which have the general form:

f(x) = ab^x

where a is the initial value, b is the growth factor, and x is the time variable.

The Given Functions

We are given two functions to consider:

A. f(x) = 1500(1.15)^x B. f(x) = 1500(1.15)^(-x)

Analyzing Function A

Let's analyze function A: f(x) = 1500(1.15)^x. In this function, the initial value (a) is 1500, which represents the initial population of bacteria. The growth factor (b) is 1.15, which represents the 115% increase in population each hour. The time variable (x) represents the number of hours elapsed.

To see if this function represents the scenario, let's calculate the population after 1 hour:

f(1) = 1500(1.15)^1 = 1500(1.15) = 1725

This means that after 1 hour, the population would increase by 1725 bacteria, making the total population 1725.

Analyzing Function B

Now, let's analyze function B: f(x) = 1500(1.15)^(-x). In this function, the initial value (a) is still 1500, but the growth factor (b) is now 1/1.15, which is less than 1. This means that the population would decrease over time, which is not consistent with the scenario.

Conclusion

Based on our analysis, function A: f(x) = 1500(1.15)^x represents the scenario of a bacterial colony with an initial population of 1500 that increases at a rate of 115% each hour. This function accurately models the exponential growth of the population over time.

Understanding Exponential Functions

Exponential functions have several key characteristics that make them useful for modeling growth and decay. Some of these characteristics include:

  • Initial Value: The initial value (a) represents the starting value of the function.
  • Growth Factor: The growth factor (b) represents the rate of increase or decrease. If b is greater than 1, the function represents growth. If b is less than 1, the function represents decay.
  • Time Variable: The time variable (x) represents the time at which the function is evaluated.

Applications of Exponential Functions

Exponential functions have many real-world applications, including:

  • Population Growth: Exponential functions can be used to model the growth of populations, such as bacteria, animals, or humans.
  • Compound Interest: Exponential functions can be used to calculate compound interest, where the interest is applied to the principal amount and the interest earned in previous periods.
  • Radioactive Decay: Exponential functions can be used to model the decay of radioactive materials, where the rate of decay is proportional to the amount of material present.

Conclusion

In conclusion, the function f(x) = 1500(1.15)^x represents the scenario of a bacterial colony with an initial population of 1500 that increases at a rate of 115% each hour. This function accurately models the exponential growth of the population over time. Exponential functions have many real-world applications, including population growth, compound interest, and radioactive decay.

References

  • [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-25.
  • [2] "Population Growth". Khan Academy. Retrieved 2023-02-25.
  • [3] "Compound Interest". Investopedia. Retrieved 2023-02-25.
  • [4] "Radioactive Decay". Physics Classroom. Retrieved 2023-02-25.

Introduction

In our previous article, we explored the concept of exponential growth and how it can be modeled using exponential functions. We also analyzed two functions to determine which one represents the scenario of a bacterial colony with an initial population of 1500 that increases at a rate of 115% each hour. In this article, we will answer some frequently asked questions (FAQs) related to exponential functions and population growth.

Q&A

Q1: What is exponential growth?

A1: Exponential growth is a type of growth where the rate of increase is proportional to the current value. In other words, the more you have, the faster it grows.

Q2: What is the formula for exponential growth?

A2: The formula for exponential growth is f(x) = ab^x, where a is the initial value, b is the growth factor, and x is the time variable.

Q3: What is the growth factor in the function f(x) = 1500(1.15)^x?

A3: The growth factor in the function f(x) = 1500(1.15)^x is 1.15, which represents a 115% increase in population each hour.

Q4: How do I calculate the population after a certain number of hours?

A4: To calculate the population after a certain number of hours, you can plug in the value of x into the function f(x) = 1500(1.15)^x. For example, to calculate the population after 2 hours, you would plug in x = 2 into the function.

Q5: What is the difference between exponential growth and linear growth?

A5: Exponential growth is a type of growth where the rate of increase is proportional to the current value, whereas linear growth is a type of growth where the rate of increase is constant. For example, if a population is growing at a rate of 10% per hour, it is an example of linear growth.

Q6: Can exponential functions be used to model population decline?

A6: Yes, exponential functions can be used to model population decline. In this case, the growth factor (b) would be less than 1, indicating a decrease in population over time.

Q7: What are some real-world applications of exponential functions?

A7: Exponential functions have many real-world applications, including population growth, compound interest, and radioactive decay.

Q8: How do I determine the initial value (a) in an exponential function?

A8: The initial value (a) is the starting value of the function. It can be determined by looking at the function and identifying the value that is being multiplied by the growth factor (b).

Q9: Can exponential functions be used to model other types of growth?

A9: Yes, exponential functions can be used to model other types of growth, such as the growth of a chemical reaction or the growth of a population of animals.

Q10: What is the significance of the time variable (x) in an exponential function?

A10: The time variable (x) represents the time at which the function is evaluated. It is an essential component of an exponential function, as it allows us to model growth or decay over time.

Conclusion

In conclusion, exponential functions are a powerful tool for modeling population growth and other types of growth. By understanding the characteristics of exponential functions and how to apply them, we can better understand and analyze real-world phenomena.

References

  • [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-25.
  • [2] "Population Growth". Khan Academy. Retrieved 2023-02-25.
  • [3] "Compound Interest". Investopedia. Retrieved 2023-02-25.
  • [4] "Radioactive Decay". Physics Classroom. Retrieved 2023-02-25.