# 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?

 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: 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) ------------------------------------------------

 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