Assignment: Analyzing Maximum Profits$\[ \begin{tabular}{|c|c|c|c|} \hline \begin{tabular}{c} Bikes \\ Produced \\ Per Day \end{tabular} & \begin{tabular}{c} Total \\ Cost \end{tabular} & \begin{tabular}{c} Total \\ Revenue \end{tabular} &
Introduction
In this assignment, we will be analyzing a business scenario where a company produces and sells bikes. The goal is to determine the maximum profit that can be achieved by the company. We will use a table to represent the data and perform calculations to find the optimal solution.
The Problem
A company produces and sells bikes. The cost of producing x bikes per day is given by the function C(x) = 100 + 20x, where C(x) is the total cost in dollars and x is the number of bikes produced per day. The revenue from selling x bikes per day is given by the function R(x) = 200x, where R(x) is the total revenue in dollars and x is the number of bikes sold per day.
We are given the following table:
| Bikes produced per day | Total cost | Total revenue |
|---|---|---|
| 0 | 100 | 0 |
| 1 | 120 | 200 |
| 2 | 140 | 400 |
| 3 | 160 | 600 |
| 4 | 180 | 800 |
| 5 | 200 | 1000 |
| 6 | 220 | 1200 |
| 7 | 240 | 1400 |
| 8 | 260 | 1600 |
| 9 | 280 | 1800 |
| 10 | 300 | 2000 |
Calculating Profit
The profit P(x) is given by the function P(x) = R(x) - C(x), where P(x) is the profit in dollars and x is the number of bikes produced per day.
We can calculate the profit for each value of x in the table:
| Bikes produced per day | Total cost | Total revenue | Profit |
|---|---|---|---|
| 0 | 100 | 0 | -100 |
| 1 | 120 | 200 | 80 |
| 2 | 140 | 400 | 260 |
| 3 | 160 | 600 | 440 |
| 4 | 180 | 800 | 620 |
| 5 | 200 | 1000 | 800 |
| 6 | 220 | 1200 | 980 |
| 7 | 240 | 1400 | 1160 |
| 8 | 260 | 1600 | 1340 |
| 9 | 280 | 1800 | 1520 |
| 10 | 300 | 2000 | 1700 |
Finding the Maximum Profit
To find the maximum profit, we need to find the value of x that maximizes the profit function P(x).
We can see from the table that the profit increases as the number of bikes produced per day increases, but at a decreasing rate. This is because the cost of producing more bikes increases, but the revenue from selling more bikes also increases.
To find the maximum profit, we can use the fact that the profit function P(x) is a quadratic function. The maximum value of a quadratic function occurs at the vertex of the parabola.
The vertex of the parabola can be found using the formula x = -b / 2a, where a and b are the coefficients of the quadratic function.
In this case, the profit function P(x) is given by the equation P(x) = -100 + 80x + 260x^2 - 400x^3 + 200x^4.
The coefficients of the quadratic function are a = 260, b = -400, and c = -100.
Using the formula x = -b / 2a, we get:
x = -(-400) / (2 * 260) x = 400 / 520 x = 0.7692
Since x must be an integer, we round up to the nearest integer to get x = 1.
However, we can see from the table that the profit is not maximized at x = 1. In fact, the profit is maximized at x = 10.
This is because the profit function P(x) is a quadratic function that opens upwards, and the vertex of the parabola is at x = 10.
Conclusion
In this assignment, we analyzed a business scenario where a company produces and sells bikes. We used a table to represent the data and performed calculations to find the optimal solution.
We calculated the profit for each value of x in the table and found that the profit is maximized at x = 10.
This means that the company should produce and sell 10 bikes per day to maximize its profit.
Discussion
The maximum profit is achieved when the company produces and sells 10 bikes per day. This is because the cost of producing more bikes increases, but the revenue from selling more bikes also increases.
However, the profit function P(x) is a quadratic function that opens upwards, and the vertex of the parabola is at x = 10. This means that the profit will decrease if the company produces and sells more than 10 bikes per day.
Therefore, the company should aim to produce and sell 10 bikes per day to maximize its profit.
Recommendations
Based on the analysis, we recommend that the company:
- Produce and sell 10 bikes per day to maximize its profit.
- Monitor the market demand and adjust the production accordingly.
- Consider increasing the price of the bikes to increase the revenue.
- Consider reducing the cost of production to increase the profit.
Limitations
The analysis is based on the assumption that the cost of production and the revenue from selling the bikes are given by the functions C(x) = 100 + 20x and R(x) = 200x, respectively.
However, in reality, the cost of production and the revenue from selling the bikes may vary depending on various factors such as the market demand, the competition, and the economic conditions.
Therefore, the analysis may not be accurate in real-world scenarios.
Future Work
In future work, we can consider the following:
- Incorporating the market demand and the competition into the analysis.
- Considering the economic conditions and the impact of inflation on the cost of production and the revenue from selling the bikes.
- Developing a more accurate model of the profit function P(x) that takes into account the various factors that affect the profit.
Introduction
In our previous article, we analyzed a business scenario where a company produces and sells bikes. We used a table to represent the data and performed calculations to find the optimal solution. In this article, we will answer some frequently asked questions related to the analysis.
Q: What is the maximum profit that can be achieved by the company?
A: The maximum profit that can be achieved by the company is $1700, which occurs when the company produces and sells 10 bikes per day.
Q: Why is the profit maximized at x = 10?
A: The profit function P(x) is a quadratic function that opens upwards, and the vertex of the parabola is at x = 10. This means that the profit will decrease if the company produces and sells more than 10 bikes per day.
Q: What is the cost of producing x bikes per day?
A: The cost of producing x bikes per day is given by the function C(x) = 100 + 20x, where C(x) is the total cost in dollars and x is the number of bikes produced per day.
Q: What is the revenue from selling x bikes per day?
A: The revenue from selling x bikes per day is given by the function R(x) = 200x, where R(x) is the total revenue in dollars and x is the number of bikes sold per day.
Q: How can the company increase its profit?
A: The company can increase its profit by producing and selling more bikes per day, but only up to a certain point. After that, the profit will decrease due to the increasing cost of production.
Q: What are the limitations of the analysis?
A: The analysis is based on the assumption that the cost of production and the revenue from selling the bikes are given by the functions C(x) = 100 + 20x and R(x) = 200x, respectively. However, in reality, the cost of production and the revenue from selling the bikes may vary depending on various factors such as the market demand, the competition, and the economic conditions.
Q: What are the recommendations for the company?
A: Based on the analysis, we recommend that the company:
- Produce and sell 10 bikes per day to maximize its profit.
- Monitor the market demand and adjust the production accordingly.
- Consider increasing the price of the bikes to increase the revenue.
- Consider reducing the cost of production to increase the profit.
Q: What are the future work recommendations?
A: In future work, we can consider the following:
- Incorporating the market demand and the competition into the analysis.
- Considering the economic conditions and the impact of inflation on the cost of production and the revenue from selling the bikes.
- Developing a more accurate model of the profit function P(x) that takes into account the various factors that affect the profit.
By considering these factors, we can develop a more accurate and comprehensive analysis of the business scenario and provide more effective recommendations to the company.
Conclusion
In this article, we answered some frequently asked questions related to the analysis of the business scenario. We provided recommendations for the company and future work recommendations to improve the analysis. We hope that this article has been helpful in understanding the analysis and its implications for the company.