Project I – Quadratic Functions

Question

Project I – Quadratic Functions Project Quadratic Functions Pro One of the many applications of quadratic functions is called the Profit Parabola. The Profit Parabola can be seen if we investigate the following scenario: The business manager of a 90 unit apartment building is trying to determine the rent to be charged. From past experience with similar buildings, when rent is set at $400, all the units are full. For every $20 increase in rent, one additional unit remains vacant. What rent should be charged for maximum total revenue? What is that maximum total revenue? 1. To help solve the above scenario, perform an internet search for Profit Parabola or Applications of Quadratic Functions. List the URL of one of the applications that you find. 2. Go to PurpleMath “Max/Min Word Problems” to see the process used for determining the quadratic function for revenues R(x) as a function of price hikes x. Use this process to determine the quadratic function that models the revenues R(x) as a function of price hikes x in the apartment building scenario above. SHOW ALL YOUR WORK! 3. What is the formula for revenues R after x $20 price hikes in the apartment building? 4. Graph the function. Clearly label the graph (desmos.com is a great online graphing resource). 5. Find the maximum revenue (or income) of the apartment building. 6. What is the rent that coincides with this maximum revenue? 7. What is the outcome of the rent hike of $20 results in 2 additional vacancies instead of 1 additional vacancy? Recalculate questions 3, 5, 6 for this new scenario. 8. Set up a similar scenario of your own invention using a business that you are interested in. Write up the scenario (problem) and the solution process involved. Find the solution to the problem you invented. Graph the function and attach it.

Dr. Jack HW Helper · Accepted Answer

The relationship between price and demand is crucial in determining revenue, and this is well illustrated through the use of quadratic functions. The problem presented in this project revolves around finding the optimal rent to maximize total revenue in a 90 unit apartment building under specific conditions, totaling a comprehensive analysis of quadratic equations and their applications in real-world scenarios. Understanding the Profit Parabola The Profit Parabola, derived from the general principles of quadratic functions, describes the relationship between revenue, price, and quantity sold. In this context, the revenue $ R $ can be expressed as a function of rent hikes. As rent increases, while revenue from occupied units rises, the total number of occupied units declines due to a rise in vacancies, which is a classic application of the quadratic function. Scenario Analysis In our scenario, all 90 units are occupied when the rent is set at $400. Upon increasing the rent by $20, one unit remains vacant. Thus, we can express the revenue function as follows: Let $ x $ be the number of $20 rent increases. The rent per apartment is given by $ 400 + 20x $, and the number of occupied units is $ 90 - x $. Therefore, the revenue function $ R(x) $ can be formulated as: R(x) = (90 - x)(400 + 20x) Expanding this, we have: R(x) = 36000 + 1800x - 20x^2 This forms a downward-opening parabola, as indicated by the negative coefficient of the $ x^2 $ term. Calculating Maximum Revenue To find the maximum revenue, we identify the vertex of the parabola represented by $ R(x) $. The x-coordinate of the vertex for a quadratic function $ ax^2 + bx + c $ is determined by the formula $ x = - \frac{b}{2a} $. Here, $ a = -20 $ and $ b = 1800 $: x = -\frac{1800}{2(-20)} = 45 This indicates 45 $20 price hikes resulting in a new rent of: 400 + 20(45) = $1400 Substituting $ x = 45 $ back into the revenue function: R(45) = (90 - 45)(400 + 20(45)) = 45(1400) = $63000 The ma

Project I – Quadratic Functions ✓ Solved

Project I – Quadratic Functions Project Quadratic Functions Pro

Paper For Above Instructions

Understanding the Profit Parabola

Scenario Analysis

Calculating Maximum Revenue

Revising for New Context

Creating a New Scenario

Graphing the Functions

Conclusion

References