Find the maximum and minimum values

Oct 13, 2025 by wbrum.org 64 views

Introduction

The average monthly temperature in a particular city can be modeled using a mathematical equation. In this article, we will explore the given model, discuss its implications, and provide insights into the underlying mathematics. The model is represented by the equation $$T(m)=4 \sin \left(\frac{\pi}{6} m\right)+2$$, where $m$ represents the months of the year.

Understanding the Model

The given model is a sinusoidal function, which means it oscillates between a maximum and minimum value. In this case, the function oscillates between $2-4$ and $2+4$ , with a period of $12$ months. This is because the coefficient of the sine function, $\frac{\pi}{6}$ , is multiplied by $m$ , which represents the months of the year. The period of a sinusoidal function is given by $\frac{2\pi}{b}$ , where $b$ is the coefficient of the sine function. In this case, the period is $\frac{2\pi}{\frac{\pi}{6}}=12$ months.

Analyzing the Model

To analyze the model, we need to find the maximum and minimum values of the function. The maximum value occurs when the sine function is equal to $1$ , which happens when $\frac{\pi}{6} m=\frac{\pi}{2}$ . Solving for $m$ , we get $m=3$ . Substituting this value into the equation, we get $T(3)=4 \sin \left(\frac{\pi}{2}\right)+2=6$ . Therefore, the maximum value of the function is $6$ degrees Celsius.

The minimum value occurs when the sine function is equal to $-1$ , which happens when $\frac{\pi}{6} m=\frac{3\pi}{2}$ . Solving for $m$ , we get $m=9$ . Substituting this value into the equation, we get $T(9)=4 \sin \left(\frac{3\pi}{2}\right)+2=-2$ . Therefore, the minimum value of the function is $-2$ degrees Celsius.

Graphical Representation

To visualize the model, we can plot the function using a graphing tool. The graph will show the oscillating behavior of the function, with the maximum and minimum values occurring at $m=3$ and $m=9$ , respectively.

Interpretation

The model provides a mathematical representation of the average monthly temperature in a particular city. The oscillating behavior of the function indicates that the temperature varies throughout the year, with the maximum and minimum values occurring in the summer and winter months, respectively.

Conclusion

In conclusion, the average monthly temperature model provides a mathematical representation of the temperature variations in a particular city. The model is a sinusoidal function, which oscillates between a maximum and minimum value. The maximum and minimum values occur at $m=3$ and $m=9$ , respectively, and the period of the function is $12$ months.

Mathematical Derivations

To derive the mathematical equations, we can use the following steps:

Find the maximum and minimum values: To find the maximum and minimum values of the function, we need to find the values of $m$ that make the sine function equal to $1$ and $-1$ , respectively.
Solve for $m$ : Solving for $m$ in the equations $\frac{\pi}{6} m=\frac{\pi}{2}$ and $\frac{\pi}{6} m=\frac{3\pi}{2}$ , we get $m=3$ and $m=9$ , respectively.
Substitute the values into the equation: Substituting the values of $m$ into the equation, we get $T(3)=4 \sin \left(\frac{\pi}{2}\right)+2=6$ and $T(9)=4 \sin \left(\frac{3\pi}{2}\right)+2=-2$ .
Find the period: The period of a sinusoidal function is given by $\frac{2\pi}{b}$ , where $b$ is the coefficient of the sine function. In this case, the period is $\frac{2\pi}{\frac{\pi}{6}}=12$ months.

Mathematical Formulas

The mathematical formulas used in this article are:

Sinusoidal function: $T(m)=4 \sin \left(\frac{\pi}{6} m\right)+2$
Period: $\frac{2\pi}{b}$
Maximum value: $T(3)=4 \sin \left(\frac{\pi}{2}\right)+2=6$
Minimum value: $T(9)=4 \sin \left(\frac{3\pi}{2}\right)+2=-2$

Mathematical Concepts

The mathematical concepts used in this article are:

Sinusoidal functions: Sinusoidal functions are periodic functions that oscillate between a maximum and minimum value.
Period: The period of a sinusoidal function is the distance between two consecutive maximum or minimum values.
Maximum and minimum values: The maximum and minimum values of a sinusoidal function occur when the sine function is equal to $1$ and $-1$ , respectively.

Mathematical Applications

The mathematical applications of this article are:

Modeling temperature variations: The model can be used to predict the average monthly temperature in a particular city.
Understanding sinusoidal functions: The article provides a mathematical representation of a sinusoidal function and its properties.
Analyzing periodic functions: The article demonstrates how to analyze periodic functions and find their maximum and minimum values.
Q&A: The Average Monthly Temperature Model =============================================

Frequently Asked Questions

Q: What is the average monthly temperature model? A: The average monthly temperature model is a mathematical equation that represents the temperature variations in a particular city. The model is given by the equation $$T(m)=4 \sin \left(\frac{\pi}{6} m\right)+2$$, where $m$ represents the months of the year.

Q: What is the period of the model? A: The period of the model is $12$ months, which means that the temperature varies between a maximum and minimum value every $12$ months.

Q: What are the maximum and minimum values of the model? A: The maximum value of the model is $6$ degrees Celsius, which occurs in the summer months, and the minimum value is $-2$ degrees Celsius, which occurs in the winter months.

Q: How can the model be used? A: The model can be used to predict the average monthly temperature in a particular city. It can also be used to understand the properties of sinusoidal functions and analyze periodic functions.

Q: What are the mathematical concepts used in the model? A: The mathematical concepts used in the model are sinusoidal functions, period, maximum and minimum values, and periodic functions.

Q: How can the model be applied in real-life situations? A: The model can be applied in real-life situations such as weather forecasting, climate modeling, and understanding the effects of global warming.

Q: What are the limitations of the model? A: The model assumes that the temperature variations are sinusoidal, which may not be the case in reality. Additionally, the model does not take into account other factors that may affect the temperature, such as humidity and wind.

Q: How can the model be improved? A: The model can be improved by incorporating more factors that affect the temperature, such as humidity and wind. Additionally, the model can be made more accurate by using more data and advanced mathematical techniques.

Q: What are the benefits of using the model? A: The benefits of using the model include being able to predict the average monthly temperature in a particular city, understanding the properties of sinusoidal functions, and analyzing periodic functions.

Q: How can the model be used in education? A: The model can be used in education to teach students about sinusoidal functions, periodic functions, and mathematical modeling. It can also be used to demonstrate the importance of mathematical concepts in real-life situations.

Q: What are the future applications of the model? A: The future applications of the model include using it to predict the effects of climate change, understanding the behavior of complex systems, and developing new mathematical techniques for modeling real-world phenomena.

Q: How can the model be used in research? A: The model can be used in research to study the behavior of complex systems, understand the effects of climate change, and develop new mathematical techniques for modeling real-world phenomena.

Q: What are the challenges of using the model? A: The challenges of using the model include understanding the mathematical concepts used in the model, collecting and analyzing data, and incorporating more factors that affect the temperature.

Q: How can the model be used in industry? A: The model can be used in industry to predict the average monthly temperature in a particular city, understand the properties of sinusoidal functions, and analyze periodic functions. It can also be used to develop new products and services related to weather forecasting and climate modeling.