Dernara Was Recently Hired As A Library Assistant At Quickwater Public Library. In The First Month, The Library Had 207 Movie Rentals. However, The Head Librarian Expects The Number Of Movie Rentals To Drop By About 10 % 10\% 10% Each Month Due To

Introduction

Dernara was recently hired as a library assistant at Quickwater Public Library. In the first month, the library had 207 movie rentals. However, the head librarian expects the number of movie rentals to drop by about $10\%$ each month due to various factors such as increased availability of streaming services and changing consumer behavior. In this article, we will explore the mathematical concept of exponential decay and use it to predict the number of movie rentals at Quickwater Public Library for the next few months.

Understanding Exponential Decay

Exponential decay is a mathematical concept that describes a situation where a quantity decreases at a rate proportional to its current value. In the case of movie rentals at Quickwater Public Library, the number of rentals is expected to decrease by $10\%$ each month. This means that the number of rentals in the second month will be $90\%$ of the number of rentals in the first month, and the number of rentals in the third month will be $90\%$ of the number of rentals in the second month, and so on.

Mathematical Model

Let's denote the number of movie rentals in the first month as $R_0$. The number of movie rentals in the second month will be $R_1 = 0.9R_0$, the number of movie rentals in the third month will be $R_2 = 0.9R_1 = 0.9^2R_0$, and so on. In general, the number of movie rentals in the $n$th month will be $R_n = 0.9^{n-1}R_0$.

Predicting the Number of Movie Rentals

Using the mathematical model above, we can predict the number of movie rentals at Quickwater Public Library for the next few months. Let's assume that the number of movie rentals in the first month is $R_0 = 207$. Then, the number of movie rentals in the second month will be $R_1 = 0.9R_0 = 0.9(207) = 186.3$, the number of movie rentals in the third month will be $R_2 = 0.9R_1 = 0.9(186.3) = 167.47$, and so on.

Calculating the Number of Movie Rentals for the Next Few Months

Conclusion

In this article, we used the mathematical concept of exponential decay to predict the number of movie rentals at Quickwater Public Library for the next few months. The results show that the number of movie rentals is expected to decrease by $10\%$ each month, resulting in a significant drop in the number of rentals over time. This prediction can be useful for the head librarian and other library staff to plan for the future and make informed decisions about the library's services and resources.

Future Research Directions

This research can be extended in several ways. For example, we can investigate the impact of other factors such as changes in consumer behavior, increased availability of streaming services, and changes in the library's services and resources on the number of movie rentals. We can also use more advanced mathematical models such as logistic growth or Gompertz growth to predict the number of movie rentals. Additionally, we can collect data on the number of movie rentals at Quickwater Public Library and compare it with the predicted values to evaluate the accuracy of the model.

References

• [1] Wikipedia. (2023). Exponential decay. Retrieved from https://en.wikipedia.org/wiki/Exponential_decay
• [2] Khan Academy. (2023). Exponential decay. Retrieved from https://www.khanacademy.org/math/differential-equations/first-order-linear-differential-equations/exponential-decay/v/exponential-decay

Appendix

The following is a Python code snippet that calculates the number of movie rentals for the next few months using the mathematical model above.

import numpy as np
def calculate_movie_rentals(R0, n):
return 0.9**(n-1)*R0
R0 = 207
n = 10
for i in range(1, n+1):
print(f"Month i} {calculate_movie_rentals(R0, i):.2f")

Introduction

In our previous article, we used the mathematical concept of exponential decay to predict the number of movie rentals at Quickwater Public Library for the next few months. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is exponential decay?

A: Exponential decay is a mathematical concept that describes a situation where a quantity decreases at a rate proportional to its current value. In the case of movie rentals at Quickwater Public Library, the number of rentals is expected to decrease by $10\%$ each month.

Q: Why does the number of movie rentals decrease by $10\%$ each month?

A: The number of movie rentals decreases by $10\%$ each month due to various factors such as increased availability of streaming services and changing consumer behavior.

Q: How can we calculate the number of movie rentals for each month?

A: We can use the mathematical model $R_n = 0.9^{n-1}R_0$ to calculate the number of movie rentals for each month, where $R_0$ is the initial number of movie rentals and $n$ is the number of months.

Q: What is the initial number of movie rentals, $R_0$?

A: The initial number of movie rentals, $R_0$, is 207.

Q: How many months will we predict the number of movie rentals for?

A: We will predict the number of movie rentals for 10 months.

Q: What is the predicted number of movie rentals for each month?

A: The predicted number of movie rentals for each month is as follows:

Q: What are some potential limitations of this model?

A: Some potential limitations of this model include:

• The model assumes that the number of movie rentals decreases by $10\%$ each month, which may not be accurate in reality.
• The model does not take into account other factors that may affect the number of movie rentals, such as changes in consumer behavior or the availability of new movies.
• The model is based on a simple exponential decay function, which may not accurately capture the complex dynamics of movie rentals.

Q: How can we improve this model?

A: We can improve this model by:

• Using more advanced mathematical models, such as logistic growth or Gompertz growth, to capture the complex dynamics of movie rentals.
• Incorporating additional data, such as changes in consumer behavior or the availability of new movies, to improve the accuracy of the model.
• Using machine learning algorithms to predict the number of movie rentals based on historical data.

Conclusion

In this article, we answered some frequently asked questions related to predicting the future of movie rentals at Quickwater Public Library. We hope that this article has provided valuable insights and information for readers who are interested in this topic.

References

• [1] Wikipedia. (2023). Exponential decay. Retrieved from https://en.wikipedia.org/wiki/Exponential_decay
• [2] Khan Academy. (2023). Exponential decay. Retrieved from https://www.khanacademy.org/math/differential-equations/first-order-linear-differential-equations/exponential-decay/v/exponential-decay

Appendix

The following is a Python code snippet that calculates the number of movie rentals for the next few months using the mathematical model above.

import numpy as np
def calculate_movie_rentals(R0, n):
return 0.9**(n-1)*R0
R0 = 207
n = 10
for i in range(1, n+1):
print(f"Month i} {calculate_movie_rentals(R0, i):.2f")

This code defines a function calculate_movie_rentals that takes the initial number of movie rentals R0 and the number of months n as input and returns the number of movie rentals for each month. The code then uses a loop to calculate and print the number of movie rentals for each month.