Situation: The IPod Touch Has Been Out For Two Years, And A Lot Of Data Has Been Collected.Relevant Relationship: There Is A Functional Relationship Between The Price Of An IPod Touch, $p$, And Weekly Demand, $s$.Below Is A Table

Introduction

The iPod Touch has been a popular portable media player for several years, and its sales data has been extensively collected. In this analysis, we will explore the relationship between the price of an iPod Touch and its weekly demand. We will use a table of data to identify the functional relationship between these two variables and make predictions about future demand based on price changes.

The Data

Price ($p$)	Weekly Demand ($s$)
99	1000
129	800
159	600
189	400
219	200

Understanding the Relationship

The relationship between the price of an iPod Touch and its weekly demand can be described using a functional relationship. This means that the demand is a function of the price, and we can write an equation to represent this relationship. Let's start by examining the data and looking for patterns.

Identifying the Pattern

As we can see from the table, as the price of the iPod Touch increases, the weekly demand decreases. This is a classic example of a negative relationship between two variables. We can represent this relationship using a linear equation of the form:

$$s = mp + b$$

where $s$ is the weekly demand, $p$ is the price, $m$ is the slope, and $b$ is the y-intercept.

Finding the Equation

To find the equation of the line, we need to determine the values of $m$ and $b$. We can do this by using the data points in the table and finding the slope and y-intercept.

Calculating the Slope

The slope of the line can be calculated using the formula:

$$m = \frac{\Delta s}{\Delta p}$$

where $\Delta s$ is the change in demand and $\Delta p$ is the change in price.

Using the data points (99, 1000) and (129, 800), we can calculate the slope as follows:

$$m = \frac{800 - 1000}{129 - 99} = \frac{-200}{30} = -\frac{20}{3}$$

Calculating the Y-Intercept

The y-intercept can be calculated using the formula:

$$b = s - mp$$

Using the data point (99, 1000), we can calculate the y-intercept as follows:

$$b = 1000 - \left(-\frac{20}{3}\right)(99) = 1000 + 660 = 1660$$

Writing the Equation

Now that we have found the values of $m$ and $b$, we can write the equation of the line as follows:

$$s = -\frac{20}{3}p + 1660$$

Interpreting the Results

The equation of the line represents the relationship between the price of an iPod Touch and its weekly demand. As the price increases, the demand decreases, and the rate of decrease is proportional to the price. This means that if the price of the iPod Touch increases by $1, the demand will decrease by $\frac{20}{3}$ units.

Making Predictions

Using the equation of the line, we can make predictions about future demand based on price changes. For example, if the price of the iPod Touch increases to $199, we can predict the demand as follows:

$$s = -\frac{20}{3}(199) + 1660 = -1320 + 1660 = 340$$

This means that if the price of the iPod Touch increases to $199, the weekly demand will be 340 units.

Conclusion

In this analysis, we have explored the relationship between the price of an iPod Touch and its weekly demand. We have identified a negative relationship between the two variables and written an equation to represent this relationship. We have also used the equation to make predictions about future demand based on price changes. This analysis demonstrates the importance of understanding the relationship between variables and using mathematical models to make predictions about future outcomes.

Future Research Directions

There are several future research directions that can be explored based on this analysis. For example, we can investigate the relationship between the price of an iPod Touch and its sales revenue. We can also explore the relationship between the price of an iPod Touch and its market share. Additionally, we can investigate the impact of other factors, such as advertising and marketing campaigns, on the demand for an iPod Touch.

References

• [1] Apple Inc. (2023). iPod Touch.
• [2] Wikipedia (2023). iPod Touch.

Appendix

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

Price ($p$)	Weekly Demand ($s$)
99	1000
129	800
159	600
189	400
219	200

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

• $s = -\frac{20}{3}p + 1660$

Introduction

In our previous article, we explored the relationship between the price of an iPod Touch and its weekly demand. We identified a negative relationship between the two variables and wrote an equation to represent this relationship. In this article, we will answer some frequently asked questions about the relationship between price and demand of the iPod Touch.

Q: What is the relationship between price and demand of the iPod Touch?

A: The relationship between price and demand of the iPod Touch is a negative relationship. As the price of the iPod Touch increases, the demand decreases.

Q: How can I use the equation to make predictions about future demand?

A: To make predictions about future demand, you can use the equation:

$$s = -\frac{20}{3}p + 1660$$

where $s$ is the weekly demand and $p$ is the price. Simply plug in the price you want to predict and solve for $s$.

Q: What is the slope of the line?

A: The slope of the line is $-\frac{20}{3}$. This means that for every $1 increase in price, the demand decreases by $\frac{20}{3}$ units.

Q: What is the y-intercept of the line?

A: The y-intercept of the line is 1660. This means that when the price is $0, the demand is 1660 units.

Q: Can I use this equation to predict demand for other products?

A: While the equation is specific to the iPod Touch, the concept of a negative relationship between price and demand can be applied to other products. However, the equation itself may not be applicable to other products.

Q: How can I collect data to create my own equation?

A: To collect data to create your own equation, you can use a variety of methods, such as:

• Conducting surveys or focus groups to gather information about consumer behavior
• Analyzing sales data or market research reports
• Using online tools or software to collect and analyze data

Q: What are some limitations of this analysis?

A: Some limitations of this analysis include:

• The data used in this analysis is limited to a specific time period and may not be representative of future trends
• The equation is based on a linear relationship, which may not accurately represent the complex relationships between price and demand
• The analysis does not take into account other factors that may influence demand, such as advertising and marketing campaigns

Q: How can I use this analysis to make business decisions?

A: This analysis can be used to inform business decisions, such as:

• Pricing strategies: By understanding the relationship between price and demand, businesses can set prices that maximize revenue
• Product development: By understanding consumer behavior and demand, businesses can develop products that meet consumer needs
• Marketing strategies: By understanding the factors that influence demand, businesses can develop marketing campaigns that effectively reach and engage with their target audience

Conclusion

In this article, we have answered some frequently asked questions about the relationship between price and demand of the iPod Touch. We have also discussed the limitations of this analysis and provided suggestions for how to use this analysis to make business decisions. By understanding the relationship between price and demand, businesses can make informed decisions that drive revenue and growth.

References

• [1] Apple Inc. (2023). iPod Touch.
• [2] Wikipedia (2023). iPod Touch.

Appendix

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

• $s = -\frac{20}{3}p + 1660$

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