Kal asked in Science & MathematicsMathematics · 5 months ago

Calculus- Economics?

A property management company manages an apartment block containing 150 units. All 150 units are rented at a monthly rate of $460 per unit and each unit costs the property management company $72.50/month for utilities and repairs. For every $25 rent increase, four fewer apartments are occupied. What rent should be charged in order to realize the most profit?

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  • 5 months ago
    Favorite Answer

    Profit is revenue minus cost.

    So when at full capacity, the revenue and costs are:

    R() = 150 * 460

    R() = 69000

    C() = 72.5 * 150

    C() = 10875

    For every $25 rent increase, 4 fewer apartments are rented. Since 150 is the max, we can't have a $25 decrease from this base.

    So if we let "x" be the number of $25 rent increases, then the rent becomes: (460 + 25x) and the number of units rented will be (150 - 4x)

    With this we can come up with updated revenue and cost functions and come up with a profit function in terms of "x":

    R(x) = (460 + 25x)(150 - 4x)

    C(x) = 72.5(150 - 4x)

    Now let's come up with the profit function and simplify:

    P(x) = R(x) - C(x)

    P(x) = (460 + 25x)(150 - 4x) - 72.5(150 - 4x)

    P(x) = 69000 - 1840x + 3750x - 100x² - 10875 + 290x

    P(x) = -100x² + 2200x + 58125

    Now that we have a profit function where "x" is the input and it's a quadratic we can solve for the maximum by solving for the zero of the first derivative:

    P'(x) = -200x + 2200

    0 = -200x + 2200

    200x = 2200

    x = 11

    So the maximum profit will be had when there are 11 increases of $25. This amount becomes:

    460 + 25x

    460 + 25 * 11

    460 + 275

    $735 per month

    And they will rent out:

    150 - 4x

    150 - 4 * 11

    150 - 44

    106 units

  • 5 months ago

    .

    Monthly rental p for each unit if n units are occupied

    ( 150, 460 )

    ( 146, 485 )

    ( p, n )

    ( p - 485 )( 146 - 150 ) = ( 485 - 460 ) ( n - 146 )

    p = ( 1397.50 - 6.25n )

    Cost of occupying n units; C = 72.50n

    Revenue from n units = np = ( 1372.50n - 6.25n² )

    Profit, P:

    P = ( 1397.50n - 6.25n² ) - 72.50n

    P = 1325n - 6.25n²

    P’(n) = 1325 - 12.5n

    1325 - 12.5n = 0

    ∴ n = 106

    p = ( 1397.50 - 6.25 * 106 )

    p = $735

    ————

  • Mike G
    Lv 7
    5 months ago

    Points given are (460,150) and (485,146)

    x = Rent, y = Units occupied

    Slope = -4/25 = -0.16

    Linear equation

    y-150 = -0.16(x-460)

    y = -0.16x + 223.6

    R = Revenue = xy = -0.16x^2+223.6x

    C = Costs = 72.5y = -11.6x+16211

    P = Profit = R-C

    P = -0.16x^2+223.6x+11.6x-16211

    P = -0.16x^2+235.2x-16211

    dP/dx = -0.32x+235.2 = 0 for maximum

    x = 235.2/0.32 = 735

    Answer $735 per month

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