The Table Below Shows The Total Cost Of Admittance To A Basketball Game, \[$ Y \$\], For \[$ X \$\] Students.$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 5 & \$25 \\ \hline 10 & \$50 \\ \hline 15 & \$75 \\ \hline 20 & \$100

Introduction

The table below presents the total cost of admittance to a basketball game for a certain number of students. The table provides a clear picture of the relationship between the number of students and the total cost of admittance. In this article, we will delve into the mathematical analysis of the table, exploring the underlying patterns and relationships that govern the data.

The Table

$x$	$y$
5	$25
10	$50
15	$75
20	$100

Observations and Patterns

At first glance, the table appears to be a simple list of numbers. However, upon closer inspection, we can observe some interesting patterns and relationships.

• The cost of admittance increases as the number of students increases.
• The cost of admittance is directly proportional to the number of students.
• The cost of admittance is $5 per student.

Mathematical Modeling

Let's assume that the cost of admittance is directly proportional to the number of students. We can model this relationship using a linear equation of the form:

$$y = mx + b$$

where $y$ is the cost of admittance, $x$ is the number of students, $m$ is the slope of the line, and $b$ is the y-intercept.

Using the data from the table, we can calculate the slope and y-intercept of the line.

• Slope ($m$): $\frac{y_2 - y_1}{x_2 - x_1} = \frac{50 - 25}{10 - 5} = \frac{25}{5} = 5$
• Y-intercept ($b$): $y = mx + b \Rightarrow b = y - mx = 25 - 5(5) = 0$

Therefore, the linear equation that models the relationship between the number of students and the cost of admittance is:

$$y = 5x$$

Graphical Representation

To visualize the relationship between the number of students and the cost of admittance, we can plot the data on a graph.

$x$	$y$
5	25
10	50
15	75
20	100

The graph shows a clear linear relationship between the number of students and the cost of admittance.

Conclusion

In conclusion, the table of basketball game admittance costs presents a clear picture of the relationship between the number of students and the total cost of admittance. Through mathematical analysis and modeling, we have identified a direct proportional relationship between the number of students and the cost of admittance. The linear equation $y = 5x$ accurately models this relationship, and the graphical representation of the data confirms the linear relationship.

Future Directions

This analysis can be extended to explore other aspects of the data, such as:

• The cost of admittance for different types of events (e.g., concerts, sports games)
• The relationship between the number of students and the cost of admittance for different age groups
• The impact of inflation on the cost of admittance over time

By exploring these and other questions, we can gain a deeper understanding of the underlying patterns and relationships that govern the data.

References

• [1] "Mathematics for Data Analysis" by John W. Tukey
• [2] "Linear Algebra and Its Applications" by Gilbert Strang

Appendix

The following is a list of the data points used in this analysis:

$x$	$y$
5	25
10	50
15	75
20	100

Introduction

In our previous article, we explored the table of basketball game admittance costs and identified a direct proportional relationship between the number of students and the cost of admittance. We also developed a linear equation $y = 5x$ to model this relationship. In this article, we will answer some frequently asked questions (FAQs) related to the table and its analysis.

Q&A

Q: What is the cost of admittance for 25 students?

A: To find the cost of admittance for 25 students, we can use the linear equation $y = 5x$. Plugging in $x = 25$, we get:

$$y = 5(25) = 125$$

Therefore, the cost of admittance for 25 students is $125.

Q: How does the cost of admittance change if the number of students increases by 10?

A: To find the change in the cost of admittance, we can use the linear equation $y = 5x$. Let's assume we have $x = 20$ students initially, and we want to find the cost of admittance for $x = 30$ students. We can plug in the values as follows:

$$y_1 = 5(20) = 100$$

$$y_2 = 5(30) = 150$$

The change in the cost of admittance is:

$$\Delta y = y_2 - y_1 = 150 - 100 = 50$$

Therefore, the cost of admittance increases by $50 if the number of students increases by 10.

Q: What is the y-intercept of the linear equation $y = 5x$?

A: The y-intercept of the linear equation $y = 5x$ is the value of $y$ when $x = 0$. Plugging in $x = 0$, we get:

$$y = 5(0) = 0$$

Therefore, the y-intercept of the linear equation $y = 5x$ is $0$.

Q: Can we use the linear equation $y = 5x$ to predict the cost of admittance for a large number of students?

A: Yes, we can use the linear equation $y = 5x$ to predict the cost of admittance for a large number of students. However, we need to be aware that the linear equation is an approximation, and the actual cost of admittance may vary depending on various factors such as inflation, event type, and age group.

Q: How does the cost of admittance compare to other types of events?

A: The cost of admittance for basketball games may be different from other types of events such as concerts or sports games. We can use the same linear equation $y = 5x$ to model the cost of admittance for other types of events, but we need to adjust the slope and y-intercept accordingly.

Q: Can we use the linear equation $y = 5x$ to predict the cost of admittance for different age groups?

A: Yes, we can use the linear equation $y = 5x$ to predict the cost of admittance for different age groups. However, we need to be aware that the linear equation is an approximation, and the actual cost of admittance may vary depending on various factors such as age group, event type, and inflation.

Conclusion

In conclusion, the table of basketball game admittance costs presents a clear picture of the relationship between the number of students and the total cost of admittance. Through mathematical analysis and modeling, we have identified a direct proportional relationship between the number of students and the cost of admittance. The linear equation $y = 5x$ accurately models this relationship, and the graphical representation of the data confirms the linear relationship. We have also answered some frequently asked questions related to the table and its analysis.

Future Directions

This analysis can be extended to explore other aspects of the data, such as:

• The cost of admittance for different types of events (e.g., concerts, sports games)
• The relationship between the number of students and the cost of admittance for different age groups
• The impact of inflation on the cost of admittance over time

By exploring these and other questions, we can gain a deeper understanding of the underlying patterns and relationships that govern the data.

References

• [1] "Mathematics for Data Analysis" by John W. Tukey
• [2] "Linear Algebra and Its Applications" by Gilbert Strang

Appendix

The following is a list of the data points used in this analysis:

$x$	$y$
5	25
10	50
15	75
20	100

This list can be used as a reference for future analyses or as a starting point for exploring other aspects of the data.