Brook West, An Apartment Complex, Has 100 Two-bedroom Units. The Monthly Profit (in Dollars) Realized From Renting X X X Apartments Is Represented By The Following Function: P ( X ) = − 9 X 2 + 1880 X − 43000 P(x) = -9x^2 + 1880x - 43000 P ( X ) = − 9 X 2 + 1880 X − 43000 (a) What Is The Actual Profit

Introduction

Brook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting $x$ apartments is represented by the following function: $P(x) = -9x^2 + 1880x - 43000$. In this article, we will explore the actual profit at Brook West apartment complex and determine the optimal number of apartments to rent in order to maximize the profit.

Understanding the Profit Function

The given profit function $P(x) = -9x^2 + 1880x - 43000$ represents the monthly profit in dollars realized from renting $x$ apartments. This function is a quadratic function, which means it has a parabolic shape. The graph of this function is a downward-facing parabola, indicating that the profit decreases as the number of apartments rented increases beyond a certain point.

Finding the Vertex of the Parabola

To find the maximum profit, we need to find the vertex of the parabola represented by the profit function. The vertex of a parabola in the form $y = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. In this case, $a = -9$ and $b = 1880$. Plugging these values into the formula, we get:

$x = -\frac{1880}{2(-9)} = \frac{1880}{18} = \frac{940}{9}$

Calculating the Maximum Profit

Now that we have found the x-coordinate of the vertex, we can plug this value back into the profit function to find the maximum profit. Substituting $x = \frac{940}{9}$ into the profit function, we get:

$P(\frac{940}{9}) = -9(\frac{940}{9})^2 + 1880(\frac{940}{9}) - 43000$

Simplifying this expression, we get:

$P(\frac{940}{9}) = -9(\frac{883600}{81}) + \frac{1767200}{9} - 43000$

$P(\frac{940}{9}) = -\frac{7948800}{81} + \frac{15801600}{81} - \frac{348000}{81}$

$P(\frac{940}{9}) = \frac{78401600}{81}$

$P(\frac{940}{9}) = 966000$

Interpreting the Results

The maximum profit of $966000$ is realized when $\frac{940}{9}$ apartments are rented. However, since the number of apartments rented cannot be a fraction, we need to round this value to the nearest whole number. Rounding $\frac{940}{9}$ to the nearest whole number, we get $104.44$, which rounds to $104$. Therefore, the actual profit at Brook West apartment complex is maximized when $104$ apartments are rented.

Conclusion

In conclusion, the actual profit at Brook West apartment complex is maximized when $104$ apartments are rented. This is determined by finding the vertex of the parabola represented by the profit function and plugging this value back into the function to find the maximum profit. The maximum profit of $966000$ is realized when $104$ apartments are rented, indicating that this is the optimal number of apartments to rent in order to maximize the profit.

Optimizing the Number of Apartments Rented

To further optimize the number of apartments rented, we can analyze the profit function and determine the range of values for which the profit is maximized. The profit function is a quadratic function, which means it has a parabolic shape. The graph of this function is a downward-facing parabola, indicating that the profit decreases as the number of apartments rented increases beyond a certain point.

Determining the Range of Values

To determine the range of values for which the profit is maximized, we can analyze the profit function and determine the values of $x$ for which the profit is greater than or equal to the maximum profit. The profit function is given by:

$P(x) = -9x^2 + 1880x - 43000$

To determine the range of values for which the profit is maximized, we can set the profit function greater than or equal to the maximum profit and solve for $x$:

$-9x^2 + 1880x - 43000 \geq 966000$

Simplifying this inequality, we get:

$-9x^2 + 1880x - 1596000 \geq 0$

Solving the Inequality

To solve this inequality, we can use the quadratic formula to find the roots of the quadratic equation:

$-9x^2 + 1880x - 1596000 = 0$

Using the quadratic formula, we get:

$x = \frac{-1880 \pm \sqrt{1880^2 - 4(-9)(-1596000)}}{2(-9)}$

Simplifying this expression, we get:

$x = \frac{-1880 \pm \sqrt{3536000 - 57120000}}{-18}$

$x = \frac{-1880 \pm \sqrt{-53560000}}{-18}$

$x = \frac{-1880 \pm 23120i}{-18}$

Since the discriminant is negative, the quadratic equation has no real roots. Therefore, the inequality has no real solutions.

Conclusion

In conclusion, the range of values for which the profit is maximized is not a real interval. The profit function is a quadratic function, which means it has a parabolic shape. The graph of this function is a downward-facing parabola, indicating that the profit decreases as the number of apartments rented increases beyond a certain point. Therefore, the optimal number of apartments to rent in order to maximize the profit is $104$.

Optimizing the Number of Apartments Rented Using Linear Programming

To further optimize the number of apartments rented, we can use linear programming to find the optimal solution. Linear programming is a method of optimization that involves finding the optimal solution to a linear objective function subject to a set of linear constraints.

Formulating the Linear Programming Problem

To formulate the linear programming problem, we need to define the objective function and the constraints. The objective function is the profit function, which is given by:

$P(x) = -9x^2 + 1880x - 43000$

The constraints are the number of apartments rented, which is a non-negative integer. Therefore, the constraints are:

$x \geq 0$

$x \in \mathbb{Z}$

Solving the Linear Programming Problem

To solve the linear programming problem, we can use the simplex method or the interior-point method. The simplex method is a popular method for solving linear programming problems, but it can be computationally expensive for large problems. The interior-point method is a more efficient method for solving linear programming problems, but it requires a good initial solution.

Conclusion

In conclusion, the optimal number of apartments to rent in order to maximize the profit is $104$. This is determined by finding the vertex of the parabola represented by the profit function and plugging this value back into the function to find the maximum profit. The maximum profit of $966000$ is realized when $104$ apartments are rented, indicating that this is the optimal number of apartments to rent in order to maximize the profit.

Future Research Directions

There are several future research directions that can be explored to further optimize the number of apartments rented. One direction is to use machine learning algorithms to predict the demand for apartments and optimize the number of apartments rented accordingly. Another direction is to use game theory to model the competition between different apartment complexes and optimize the number of apartments rented accordingly.

Conclusion

In conclusion, the optimal number of apartments to rent in order to maximize the profit is $104$. This is determined by finding the vertex of the parabola represented by the profit function and plugging this value back into the function to find the maximum profit. The maximum profit of $966000$ is realized when $104$ apartments are rented, indicating that this is the optimal number of apartments to rent in order to maximize the profit.

Q: What is the profit function for Brook West apartment complex?

A: The profit function for Brook West apartment complex is given by $P(x) = -9x^2 + 1880x - 43000$, where $x$ is the number of apartments rented.

Q: What is the maximum profit that can be realized at Brook West apartment complex?

A: The maximum profit that can be realized at Brook West apartment complex is $966000$, which is realized when $104$ apartments are rented.

Q: How is the optimal number of apartments to rent determined?

A: The optimal number of apartments to rent is determined by finding the vertex of the parabola represented by the profit function and plugging this value back into the function to find the maximum profit.

Q: What is the significance of the vertex of the parabola in determining the optimal number of apartments to rent?

A: The vertex of the parabola represents the point at which the profit is maximized. Plugging this value back into the profit function gives the maximum profit that can be realized.

Q: Can the number of apartments rented be a fraction?

A: No, the number of apartments rented cannot be a fraction. Therefore, the optimal number of apartments to rent must be rounded to the nearest whole number.

Q: How is the range of values for which the profit is maximized determined?

A: The range of values for which the profit is maximized is determined by setting the profit function greater than or equal to the maximum profit and solving for $x$.

Q: Can the quadratic equation have real roots?

A: No, the quadratic equation has no real roots, which means that the inequality has no real solutions.

Q: What is the significance of the linear programming problem in determining the optimal number of apartments to rent?

A: The linear programming problem is used to find the optimal solution to the profit function subject to the constraints of the number of apartments rented.

Q: What are the constraints of the linear programming problem?

A: The constraints of the linear programming problem are the number of apartments rented, which is a non-negative integer.

Q: Can the linear programming problem be solved using the simplex method or the interior-point method?

A: Yes, the linear programming problem can be solved using the simplex method or the interior-point method.

Q: What are the future research directions for optimizing the number of apartments rented?

A: Some future research directions include using machine learning algorithms to predict the demand for apartments and optimize the number of apartments rented accordingly, and using game theory to model the competition between different apartment complexes and optimize the number of apartments rented accordingly.

Q: What is the conclusion of the study on maximizing profit at Brook West apartment complex?

A: The conclusion of the study is that the optimal number of apartments to rent in order to maximize the profit is $104$, which is determined by finding the vertex of the parabola represented by the profit function and plugging this value back into the function to find the maximum profit.

Q: What are the implications of the study on maximizing profit at Brook West apartment complex?

A: The implications of the study are that the optimal number of apartments to rent in order to maximize the profit is $104$, which can be used by the management of Brook West apartment complex to make informed decisions about the number of apartments to rent.

Q: What are the limitations of the study on maximizing profit at Brook West apartment complex?

A: The limitations of the study are that it assumes a quadratic profit function and does not take into account other factors that may affect the profit, such as the cost of renting apartments and the demand for apartments.

Q: What are the future research directions for the study on maximizing profit at Brook West apartment complex?

A: Some future research directions include using machine learning algorithms to predict the demand for apartments and optimize the number of apartments rented accordingly, and using game theory to model the competition between different apartment complexes and optimize the number of apartments rented accordingly.