Sophia Owns A Clothing Store That Sells Graphic T-shirts. { N $}$ Is The Number Of Shirts She Sells Each Month. The Revenue Function Of Her Store Is $ R = 21n $. The Cost Function Of Her Store Is $ C = 16n + 750 $. Using
Maximizing Profit in a Clothing Store: A Case Study of Sophia's Graphic T-Shirts
Introduction
As a business owner, Sophia's clothing store is her primary source of income. She sells graphic T-shirts, which are a popular item among her target audience. The success of her store depends on various factors, including the number of shirts she sells each month, the revenue generated, and the costs associated with running the business. In this article, we will explore the revenue and cost functions of Sophia's store and discuss how to maximize her profit.
Revenue Function
The revenue function of Sophia's store is given by the equation $ r = 21n $, where $ n $ is the number of shirts she sells each month. This means that for every shirt sold, Sophia generates a revenue of $ 21. The revenue function is a linear equation, indicating that the revenue increases proportionally with the number of shirts sold.
Revenue Function: r = 21n
Cost Function
The cost function of Sophia's store is given by the equation $ C = 16n + 750 $, where $ n $ is the number of shirts she sells each month. This means that for every shirt sold, Sophia incurs a cost of $ 16, in addition to a fixed cost of $ 750. The cost function is also a linear equation, indicating that the cost increases proportionally with the number of shirts sold.
Cost Function: C = 16n + 750
Profit Function
The profit function of Sophia's store is the difference between the revenue and the cost. It is given by the equation $ P = r - C = 21n - (16n + 750) = 5n - 750 $. This means that for every shirt sold, Sophia's profit increases by $ 5, minus a fixed cost of $ 750.
Profit Function: P = 5n - 750
Maximizing Profit
To maximize Sophia's profit, we need to find the optimal number of shirts to sell each month. This can be done by setting the profit function equal to zero and solving for $ n $. However, since the profit function is a linear equation, it is not possible to find a maximum value for $ n $. Instead, we can analyze the profit function to determine the conditions under which Sophia's profit is maximized.
Maximizing Profit: P = 5n - 750
Break-Even Analysis
To determine the break-even point, we need to find the number of shirts that Sophia needs to sell to cover her fixed costs. This can be done by setting the profit function equal to zero and solving for $ n $. The break-even point is given by the equation $ 5n - 750 = 0 $, which implies that $ n = 150 $. This means that Sophia needs to sell at least 150 shirts each month to cover her fixed costs.
Break-Even Analysis: n = 150
Sensitivity Analysis
To analyze the sensitivity of Sophia's profit to changes in the number of shirts sold, we can use the profit function. For example, if Sophia sells 100 shirts each month, her profit is $ P = 5(100) - 750 = 250 $. If she sells 200 shirts each month, her profit is $ P = 5(200) - 750 = 500 $. This shows that Sophia's profit increases by $ 250 for every additional 100 shirts sold.
Sensitivity Analysis: P = 5n - 750
Conclusion
In conclusion, Sophia's clothing store is a successful business that generates revenue from the sale of graphic T-shirts. The revenue and cost functions of her store are linear equations, indicating that the revenue and cost increase proportionally with the number of shirts sold. The profit function is also a linear equation, indicating that the profit increases proportionally with the number of shirts sold. To maximize Sophia's profit, we need to find the optimal number of shirts to sell each month. This can be done by analyzing the profit function and determining the conditions under which Sophia's profit is maximized.
Recommendations
Based on the analysis of Sophia's store, we recommend the following:
- Increase the number of shirts sold: To maximize Sophia's profit, we recommend increasing the number of shirts sold each month. This can be done by implementing marketing strategies to attract more customers and increasing the production of graphic T-shirts.
- Reduce fixed costs: To reduce Sophia's fixed costs, we recommend renegotiating the lease agreement with the landlord or finding a more affordable location for the store.
- Improve product quality: To improve the quality of the graphic T-shirts, we recommend investing in better materials and manufacturing processes.
By implementing these recommendations, Sophia can maximize her profit and ensure the long-term success of her clothing store.
Sophia's Clothing Store: A Q&A Guide to Maximizing Profit
Introduction
In our previous article, we explored the revenue and cost functions of Sophia's clothing store and discussed how to maximize her profit. In this article, we will answer some frequently asked questions about Sophia's store and provide additional insights into the business.
Q&A
Q: What is the revenue function of Sophia's store?
A: The revenue function of Sophia's store is given by the equation $ r = 21n $, where $ n $ is the number of shirts she sells each month.
Q: What is the cost function of Sophia's store?
A: The cost function of Sophia's store is given by the equation $ C = 16n + 750 $, where $ n $ is the number of shirts she sells each month.
Q: What is the profit function of Sophia's store?
A: The profit function of Sophia's store is given by the equation $ P = 5n - 750 $, where $ n $ is the number of shirts she sells each month.
Q: How can Sophia maximize her profit?
A: To maximize Sophia's profit, we recommend increasing the number of shirts sold each month, reducing fixed costs, and improving product quality.
Q: What is the break-even point for Sophia's store?
A: The break-even point for Sophia's store is given by the equation $ 5n - 750 = 0 $, which implies that $ n = 150 $. This means that Sophia needs to sell at least 150 shirts each month to cover her fixed costs.
Q: How does Sophia's profit change with the number of shirts sold?
A: Sophia's profit increases by $ 250 for every additional 100 shirts sold. This can be seen from the profit function $ P = 5n - 750 $.
Q: What are some potential risks for Sophia's store?
A: Some potential risks for Sophia's store include:
- Competition: Sophia's store may face competition from other clothing stores in the area.
- Economic downturn: An economic downturn may reduce demand for graphic T-shirts and negatively impact Sophia's store.
- Supply chain disruptions: Disruptions to the supply chain may impact Sophia's ability to produce and sell graphic T-shirts.
Conclusion
In conclusion, Sophia's clothing store is a successful business that generates revenue from the sale of graphic T-shirts. By understanding the revenue and cost functions of her store, Sophia can make informed decisions to maximize her profit. We hope that this Q&A guide has provided additional insights into the business and has been helpful to readers.
Recommendations
Based on the analysis of Sophia's store, we recommend the following:
- Monitor competition: Sophia should monitor the competition in the area and adjust her marketing strategies accordingly.
- Diversify products: Sophia should consider diversifying her products to reduce dependence on graphic T-shirts.
- Improve supply chain management: Sophia should improve her supply chain management to reduce the risk of disruptions.
By implementing these recommendations, Sophia can minimize the risks associated with her store and ensure its long-term success.
Additional Resources
For additional resources on Sophia's store, including financial statements and marketing strategies, please visit our website.
Contact Us
If you have any questions or would like to learn more about Sophia's store, please contact us at info@sophiasstore.com.