algebra

David makes and sells chairs. The function p(x)=-10x^2 + 100x - 210 indicates how much profit he makes in a month if he sells the chairs for 10 - x dollars each.

What should David charge per chair to make the maximum profit, and what is the maximum profit he can make in a month?

a) $40 at $5 per chair**
b) $50 at $7 per chair
c) $40 at $7 per chair
d) $50 at $5 per chair
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I need help on this, I do not know where to start. After looking on a graphing calculator, I think the answer may be a....

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  1. you want the max value of -10x^2 + 100x - 210
    = -10(x^2 - 10x + ... - .....) - 210
    I am completing the square
    = -10(x^2 - 10x + 25-25) - 210
    = -10( (x-5)^2 - 25) - 210
    = -10(x-5)^2 + 250 - 210
    = -10(x-5)^2 + 40

    Max is 40 when x = 5

    how does that fit into the given answers?
    remember you want to calculate 10-x for the selling price of a chair.

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  2. a x² + bx + c

    is called a quadratic function.

    The graph of a quadratic function is a parabola.

    The lowest or the highest point on a parabola is called the vertex.

    For:

    a > 0

    parabola is opens up and have minimum value in vertex.

    For:

    a < 0

    parabola is opens down and have maximum value in vertex.

    The vertex has the x - coordinate:

    x = - b / 2 a

    In this case:

    p(x) = - 10 x² + 100 x - 210

    a = - 10 , b = 100 , c = - 210

    Vertex x - coordinate:

    x = - b / 2 a = - 100 / 2 ∙ ( - 10 ) = - 100 / - 20 = 5

    p(x) = - 10 x² + 100 x - 210

    For x = 5

    p(5) = - 10 ∙ 5² + 100 ∙ 5 - 210 =

    - 10 ∙ 25 + 500 - 210 =

    - 250 + 500 - 210 = 40

    Maximum profit is at the vertex of the parabola :

    x = 5 , p = 40

    The price is 10 - x = $10 - $ 5 = $5

    so:

    $40 at $5 per chair

    Answer a) is correct.

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  3. Thank you so much, guys!! You have really made me understand this now. I appreciate it!
    Have a great night! :)

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