Grade 12 - The manager knows 100-unit apartment complex will be occupied if the rent is $400 per month

A market survey suggests that, on the average, one additional unit will remain vacant for each $5 increase in rent. What rent should the manager charge to maximize revenue?
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  • JuliaZ 7 months

    you know that at $400, there is 100 unit rented out,
    so Revenue = 100 * 400.
    you also know as rent increase by 5, unit decrease by 1.
    we can rewrite the equation as,
    Revenue = (100 – n)(500 + 5*n)
    n for each $5 increase.
    Expanding the equation gives
    R = -5x^2 + 100n + 4000

    we can use this equation to find the derivative of each term
    so derivative of -5x ^ 2 is -5(2)x
    derivative of 100n is 100
    derivative of 4000 is 0 because it is a constant

    now combining the derivatives, the derivative of revenue
    is -10n + 100. when at the maximum revenue, the tangent must be 0. so
    -10n + 100 = 0
    n = 10.
    n is the number of $5 increases.
    so manager should charge 400 + 5(10) = 450 to maximize revenue.