logo for solving-math-problems.com
leftimage for solving-math-problems.com

Quadratic Profit Function - Real Estate










































Quadratic Profit Function


Old Bib Real Estate has a 100 unit apartment and plans to rent out the apartment. The monthly profit generated by renting out x units of the apartment is given by P(x)=-10x²+1760x-50000 . How many units should be rented out to maximize the profit?

Comments for Quadratic Profit Function - Real Estate

Click here to add your own comments

Sep 09, 2012
Quadratic Profit Function
by: Staff


Answer:




Part I


The monthly profit function is: P(x)=-10x^2+1760x-50000 .

A graph of the profit function is shown below:



 Graph of Quadratic Profit Function P(x) = -10x² + 1760x - 50000





As you can see, finding the maximum profit is equivalent to finding the vertex of the parabola.


There are three good ways of finding the vertex of the parabola.

   1. Using the first term of the quadratic formula

         the general form of a parabola is:

           y = ax² + bx + x

         the quadratic formula is:

           x = (-b/2a) ± [√(b² - 4ac)/2a]

         the x coordinate of the vertex of the parabola is:

           x_vertex = (-b/2a)

         for your equation: P(x) = -10x² + 1760x - 50000

         a = -10

         b = 1760

         x_vertex = -(1760)/[2*(-10)]

         x_vertex = -1760/(-20)

         x_vertex = 88


         now that you know the x coordinate of the vertex, substitute this value in the original equation to compute y

         P(x) = -10x² + 1760x - 50000

         x = 88

         P(x) = -10*88² + 1760*88 - 50000

         P(x) = -77440 + 154880 - 50000

         P(x) = 27440

         the coordinates of the vertex in x,y format are: (88, 27440)

------------------------------------------------

Sep 09, 2012
Quadratic Profit Function
by: Staff


------------------------------------------------

Part II


   2. Rewrite the quadratic profit function in vertex format


         P(x) = -10x² + 1760x - 50000

         P(x) = (-10x² + 1760x) - 50000

         P(x) = -10(x² - 176x) - 50000

            
176 / 2 = 88

            
88² = 7744

         P(x) = -10(x² - 176x + 7744 - 7744) - 50000

         P(x) = -10(x² - 176x + 7744) - (-10)*7744 - 50000

         P(x) = -10(x² - 176x + 7744) + 77440 - 50000

         P(x) = -10(x - 88)² + 27440


      The vertex form of the quadratic equation shows you exactly what the coordinates of the vertex are:

         the coordinates of the vertex in x,y format are: (88, 27440)
         (this is exactly the same answer calculated using the first term of the quadratic formula)



   3. Take the derivative of the quadratic profit function


         Profit Function: P(x) = -10x² + 1760x – 50000

         take the derivative: d/dx[-10x² + 1760x - 50000]

         d/dx = -20x + 1760

         set the derivative equal to 0: 0 = -20x + 1760

         solve for x: 0 + 20x = -20x + 20x + 1760

         0 + 20x = 0 + 1760

         20x = 1760

         20x / 20 = 1760 / 20

         x * (20 / 20) = 1760 / 20

         x * (1) = 1760 / 20

         x = 1760 / 20

         x = 88

------------------------------------------------

Sep 09, 2012
Quadratic Profit Function
by: Staff


------------------------------------------------

Part III

         now that you know the x coordinate of the vertex, substitute this value in the original equation to compute y

         P(x) = -10x² + 1760x - 50000

         x = 88

         P(x) = -10*88² + 1760*88 - 50000

         P(x) = -77440 + 154880 - 50000

         P(x) = 27440


         the coordinates of the vertex in x,y format are: (88, 27440)
         (these are the same coordinates which were calculated in 1 and 2.)




Final Answer

    How many units should be rented out to maximize the profit?

         88 units




Thanks for writing.

Staff
www.solving-math-problems.com



Click here to add your own comments

Join in and write your own page! It's easy to do. How? Simply click here to return to Math Questions & Comments - 01.



Copyright © 2008-2015. All rights reserved. Solving-Math-Problems.com