Kal asked in Science & MathematicsMathematics · 5 months ago

Need Help Quickly PLEASE in Calculus- Optimization in 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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  • Best answer

    Income - Cost = Profit

    (460 + 25 * x) * (150 - 4 * x) - 72.50 * 150 = P

    P = 460 * 150 + 25 * 150 * x - 460 * 4 * x - 100 * x^2 - 7250

    P = 46 * 15 * 100 - 7250 + 3750x - 1840x - 100x^2

    P = 69000 - 7250 + 1910x - 100x^2

    dP/dx = 0 + 1910 - 200x

    dP/dx = 0

    0 = 1910 - 200x

    0 = 191 - 20x

    20x = 191

    x = 191/20

    x = 95.5/10

    x = 9.55

    x = 9 or x = 10. Maybe even x = 9.5. By the way, I'm assuming that the utilities and repairs go for each unit, whether or not they're occupied. We can change things later if we wish

    x = 9.5

    P = 69000 - 7250 + 1910x - 100x^2

    P = 61750 + 1910 * 9.5 - 100 * 9.5^2

    P = 61750 + 191 * 95 - 100 * (19/2)^2

    P = 61750 + 191 * (100 - 5) - 100 * (361/4)

    P = 61750 + 19100 - 955 - 36100/4

    P = 80850 - 955 - 9025

    P = 71825 - 955

    P = 72000 - 175 - 955

    P = 71045 - 175

    P = 71000 - 135

    P = 70865

    460 + 25 * 9.5 = 460 + 250 - 12.5 = 710 - 12.5 = 697.5

    Assuming that utilities and repairs only apply to apartments that are rented

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

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

    P = (387.5 + 25x) * 2 * (75 - 2x)

    P = (775 + 50x) * (75 - 2x)

    P = 25 * (31 + 2x) * (75 - 2x)

    dP/dx = 25 * (31 + 2x) * (-2) + 25 * (75 - 2x) * 2

    dP/dx = 0

    0 = 25 * 2 * (75 - 2x - 31 - 2x)

    0 = 75 - 31 - 4x

    0 = 44 - 4x

    4x = 44

    x = 11

    460 + 25 * 11

    460 + 275

    735

    Rent should be $735 per month. Considering that the numbers work nicer with this assumption, I'm going to go ahead and say that this is what they wanted.

    P = 25 * (31 + 2x) * (75 - 2x)

    P = 25 * (31 + 22) * (75 - 22)

    P = 25 * 53 * 53

    P = 25 * (52 + 1)^2

    P = 25 * (52^2 + 2 * 52 + 1)

    P = 25 * 52 * 52 + 25 * 2 * 52 + 25 * 1

    P = 25 * 4 * 13 * 52 + 50 * 52 + 25

    P = 100 * 13 * 13 * 4 + 100 * 26 + 25

    P = 100 * 169 * 4 + 2600 + 25

    P = 100 * 676 + 2625

    P = 67600 + 2625

    P = 70225

  • 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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