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Quadratic Function - Maximize Rental Profit











































Maximize Rental Profit

   • 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) = -11x² + 1936x - 52000.

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

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Sep 01, 2012
Maximize Rental Profit
by: Staff


Answer:

Part I


Note: the monthly profit function submitted in the problem statement did not display. I can only guess what the function is. I added my own function: P(x) = -11x² + 1936x - 52000. If this is not the function which should be used, the solution using this function will, at least, demonstrate how to solve this type of problem.


A graph of the profit function is shown below:


Graph of Quadratic Profit Function P(x) = -11x² + 1936x - 52000





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) = -11x² + 1936x - 52000

         a = -11

         b = 1936

         x_vertex = -(1936)/[2*(-11)]

         x_vertex = -1936/(-22)

         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) = -11x² + 1936x - 52000

         x = 88

         P(x) = -11*88² + 1936*88 - 52000

         P(x) = -85184 + 170368 - 52000

         P(x) = 33184

         the coordinates of the vertex in x,y format are: (88, 33184)

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Sep 01, 2012
Maximize Rental Profit
by: Staff

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

Part II


   2. Rewrite the quadratic profit function in vertex format


         P(x) = -11x² + 1936x - 52000

         P(x) = (-11x² + 1936x) - 52000

         P(x) = -11(x² - 176x + 7744 - 7744) - 52000

         P(x) = -11(x² - 176x + 7744) - (-11)*7744 – 52000

         P(x) = -11(x² - 176x + 7744) + 85184 - 52000

         P(x) = -11(x - 88)² + 33184


      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, 33184)
         (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) = -11x² + 1936x - 52000

         take the derivative: d/dx[-11x² + 1936x - 52000]

         d/dx = -22x² + 1936

         set the derivative equal to 0: 0 = -22x + 1936

         solve for x: 0 = -22x + 1936

         0 + 22x = -22x + 22x + 1936

         0 + 22x = 0 + 1936

         22x = 1936

         22x / 22 = 1936 / 22

         x * (22 / 22) = 1936 / 22

         x * (1) = 1936 / 22

         x = 1936 / 22

         x = 88

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

Sep 01, 2012
Maximize Rental Profit
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) = -11x² + 1936x - 52000

         x = 88

         P(x) = -11*88² + 1936*88 - 52000

         P(x) = -85184 + 170368 - 52000

         P(x) = 33184


         the coordinates of the vertex in x,y format are: (88, 33184)
         (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



Sep 05, 2012
A bit confused
by: Stef

Dear Staff,

Why the Part II working from this

P(x) = -11(x² - 176x + 7744) + 85184 - 52000 become

P(x) = -11(x - 88)² - 33184 and not

P(x) = -11(x - 88)² + 33184? I'm a bit lost here cos 85184 - 52000 should be positive. Can you please help me on this puzzle?


Rgds

Sep 05, 2012
Quadratic Function - Maximize Rental Profit
by: Staff

Hello Stef,

You are right. It should be +33184.

Thanks for pointing that out.


Staff
www.solving-math-problems.com

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