The Profit Of A Cell-phone Manufacturer Is Found By The Function $y = -2x^2 + 108x + 75$, Where $x$ Is The Cost Of The Cell Phone. At What Price Should The Manufacturer Sell The Phone To Maximize Its Profits? What Will The Maximum

Introduction

In the competitive world of cell-phone manufacturing, companies are constantly seeking ways to maximize their profits while maintaining a competitive edge in the market. One key factor in determining a company's profitability is the price at which they sell their products. In this article, we will explore how to use mathematical optimization techniques to determine the optimal price at which a cell-phone manufacturer should sell their phones to maximize their profits.

The Profit Function

The profit of a cell-phone manufacturer is given by the function $y = -2x^2 + 108x + 75$, where $x$ is the cost of the cell phone. This function represents the relationship between the cost of the cell phone and the resulting profit. To maximize profits, we need to find the value of $x$ that maximizes the function.

Understanding the Function

Before we can find the maximum value of the function, we need to understand its behavior. The function is a quadratic function, which means it has a parabolic shape. The coefficient of the $x^2$ term is negative, indicating that the parabola opens downward. This means that the function has a maximum value at some point, which we need to find.

Finding the Maximum Value

To find the maximum value of the function, we need to find the vertex of the parabola. The vertex is the point at which the parabola changes direction, and it is the maximum or minimum point of the function. In this case, we are interested in the maximum point.

The Vertex Formula

The vertex of a parabola can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the $x^2$ and $x$ terms, respectively. In this case, $a = -2$ and $b = 108$. Plugging these values into the formula, we get:

$x = -\frac{108}{2(-2)}$ $x = -\frac{108}{-4}$ $x = 27$

The Maximum Value

Now that we have found the value of $x$ at which the function has a maximum value, we can plug this value back into the original function to find the maximum value of the function. Plugging $x = 27$ into the function, we get:

$y = -2(27)^2 + 108(27) + 75$ $y = -2(729) + 2916 + 75$ $y = -1458 + 2916 + 75$ $y = 1533$

Conclusion

In conclusion, the cell-phone manufacturer should sell their phones at a price of $x = 27$ to maximize their profits. This will result in a maximum profit of $y = 1533$. By using mathematical optimization techniques, we have been able to determine the optimal price at which the manufacturer should sell their phones to maximize their profits.

Maximizing Profits through Optimization

Optimization is a powerful tool that can be used to maximize profits in a variety of industries. By using mathematical techniques such as quadratic functions and vertex formulas, we can determine the optimal price at which a product should be sold to maximize profits. This can be particularly useful in industries such as manufacturing, where the cost of production is a key factor in determining profitability.

The Importance of Optimization

Optimization is an important tool in many industries, including manufacturing, finance, and logistics. By using optimization techniques, companies can maximize their profits, reduce costs, and improve efficiency. In the case of the cell-phone manufacturer, optimization has allowed us to determine the optimal price at which the manufacturer should sell their phones to maximize their profits.

Real-World Applications

Optimization has many real-world applications in a variety of industries. For example, in finance, optimization can be used to determine the optimal investment portfolio to maximize returns. In logistics, optimization can be used to determine the most efficient route for delivering goods. In manufacturing, optimization can be used to determine the optimal production schedule to maximize efficiency.

Conclusion

In conclusion, optimization is a powerful tool that can be used to maximize profits in a variety of industries. By using mathematical techniques such as quadratic functions and vertex formulas, we can determine the optimal price at which a product should be sold to maximize profits. This can be particularly useful in industries such as manufacturing, where the cost of production is a key factor in determining profitability.

References

• [1] "Quadratic Functions." Math Open Reference, mathopenref.com/quadratic.html.
• [2] "Vertex Formula." Math Is Fun, mathisfun.com/algebra/vertex-formula.html.
• [3] "Optimization." Investopedia, investopedia.com/terms/o/optimization.asp.

Further Reading

• [1] "Quadratic Functions and Optimization." Khan Academy, khanacademy.org/math/algebra/quadratic-functions-and-optimization.
• [2] "Optimization Techniques." Coursera, coursera.org/specializations/optimization.
• [3] "Mathematical Optimization." Wikipedia, en.wikipedia.org/wiki/Mathematical_optimization.
  

Introduction

In our previous article, we explored how to use mathematical optimization techniques to determine the optimal price at which a cell-phone manufacturer should sell their phones to maximize their profits. We used the function $y = -2x^2 + 108x + 75$ to find the maximum value of the function, which resulted in a maximum profit of $y = 1533$ at a price of $x = 27$. In this article, we will answer some of the most frequently asked questions about the topic.

Q&A

Q: What is the purpose of using optimization techniques in cell-phone manufacturing?

A: The purpose of using optimization techniques in cell-phone manufacturing is to determine the optimal price at which the manufacturer should sell their phones to maximize their profits. By using mathematical techniques such as quadratic functions and vertex formulas, we can determine the optimal price at which the manufacturer should sell their phones to maximize their profits.

Q: How do I use the vertex formula to find the maximum value of the function?

A: To use the vertex formula, you need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$. In this case, $a = -2$ and $b = 108$. Plugging these values into the formula, we get $x = -\frac{108}{2(-2)} = 27$.

Q: What is the maximum value of the function?

A: The maximum value of the function is $y = 1533$, which occurs at a price of $x = 27$.

Q: How do I determine the optimal price at which the manufacturer should sell their phones?

A: To determine the optimal price at which the manufacturer should sell their phones, you need to use the vertex formula to find the maximum value of the function. In this case, the maximum value of the function is $y = 1533$, which occurs at a price of $x = 27$.

Q: What are some real-world applications of optimization techniques in cell-phone manufacturing?

A: Some real-world applications of optimization techniques in cell-phone manufacturing include:

• Determining the optimal production schedule to maximize efficiency
• Determining the optimal price at which the manufacturer should sell their phones to maximize profits
• Determining the optimal investment portfolio to maximize returns

Q: How do I use optimization techniques to determine the optimal price at which the manufacturer should sell their phones?

A: To use optimization techniques to determine the optimal price at which the manufacturer should sell their phones, you need to use the vertex formula to find the maximum value of the function. In this case, the maximum value of the function is $y = 1533$, which occurs at a price of $x = 27$.

Q: What are some common mistakes to avoid when using optimization techniques in cell-phone manufacturing?

A: Some common mistakes to avoid when using optimization techniques in cell-phone manufacturing include:

• Not using the correct values of $a$ and $b$ in the vertex formula
• Not plugging in the correct values of $a$ and $b$ into the formula
• Not using the correct formula to find the maximum value of the function

Conclusion

In conclusion, optimization techniques can be used to determine the optimal price at which a cell-phone manufacturer should sell their phones to maximize their profits. By using mathematical techniques such as quadratic functions and vertex formulas, we can determine the optimal price at which the manufacturer should sell their phones to maximize their profits. We hope that this article has been helpful in answering some of the most frequently asked questions about the topic.

Further Reading

• [1] "Quadratic Functions and Optimization." Khan Academy, khanacademy.org/math/algebra/quadratic-functions-and-optimization.
• [2] "Optimization Techniques." Coursera, coursera.org/specializations/optimization.
• [3] "Mathematical Optimization." Wikipedia, en.wikipedia.org/wiki/Mathematical_optimization.

References

• [1] "Quadratic Functions." Math Open Reference, mathopenref.com/quadratic.html.
• [2] "Vertex Formula." Math Is Fun, mathisfun.com/algebra/vertex-formula.html.
• [3] "Optimization." Investopedia, investopedia.com/terms/o/optimization.asp.