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12 August, 08:20

The function f (x) = - x2 + 48x - 432 models the profit, in dollars, a game developer makes for selling a new game, where x is the number of games sold and f (x) is the amount of profit. Part A: Determine the vertex. What does this calculation mean in the context of the problem? Part B: Determine the x-intercepts. What do these values mean in the context of the problem?

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  1. 12 August, 11:06
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    f (x) = - x^2 + 48x - 432

    Part A

    Find the zeros:

    -x^2 + 48x - 432 = 0

    x^2 - 48x + 432 = 0

    Factor

    (x - 36) (x - 12) = 0

    Two solutions: x = 36 and x = 12

    The vertex is in x = (36 + 12) / 2 = 24

    f (24) = - (24) ^2 + 48 (24) - 432 = 144

    The vertex is (24, 144).

    Given that the coefficient of x^2 is negative, the parabole opens downward, that mean that the vertex is a maximum value.

    In the context of the problem that means that the maximum profit is 144 and is obtainded when the number of games sold is 24.

    Part B.

    The x-intercepts are the values of x when y=0. Which we calculated above.

    They are x = 24 and x = 36.

    Those values are the number of games for which the profit is zero.
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