Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

confusing math practice sat question?

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  • 1 month ago
    Favorite Answer

    The function P(x) = -2x² + 20x - 48 represents a parabola.

    Since the leading coefficient (-2) is negative, we know we have a downward facing parabola, so the vertex will be a maximum.

    In general, the x-coordinate of the vertex of a parabola (ax² + bx + c) can be found using:

    x = -b/(2a)

    In your case:

    x = -20/(2 * -2)

    x = -20/-4

    x = 5

    Plugging in x = 5, we get:

    P(5) = -2(5²) + 20(5) - 48

    P(5) = -50 + 100 - 48

    P(5) = 2

    That's the company's profit in *thousands* of dollars.

    Answer:B) $2,000

    Another way to get that result is to factor the quadratic:

    P(X) = -2(x² - 10x + 24)

    P(X) = -2(x - 4)(x - 6)

    The two roots are x = 4 and x = 6. We know the line of symmetry is halfway between these two roots, so we can mentally figure out it's 5. (Alternatively, take the average of 4 and 6 --> (4+6)/2 = 5)

    Then calculate P(5) as before.

    Another method, if you know calculus is to take the derivative.

    P'(x) = -4x + 20

    Set it to zero:

    -4x + 20 = 0

    -4x = -20

    x = -20/-4

    x = 5

    However you get to x=5, calculate P(5) = 2 and multiply it by 1000.

    Answer:B) $2,000

  • 1 month ago

    P(x) = -2x² + 20x - 48

    or, P(x) = -2(x² - 10x) - 48

    so, P(x) = -2(x - 5)² - 48 + 50

    Hence, P(x) = -2(x - 5)² + 2    

    We can see that x value of x = 5 gives P(5) = 2

    Therefore, selling an item for $5.00 will yield a maximum profit of $2000.

    :)>

  • Philip
    Lv 6
    1 month ago

    P(x) = -2x^2 +20x -48.;

    P'(x) = -4x+20, = 0 at [x,P(x)] = (5,2).;

    P''(x) = -4, < 0----> any extremum is a maximum.;

    Clearly, max profit occurs where P(x) = 2 which corresponds to choice B)

  • 1 month ago

    You don't need calculus for this one.  The polynomial factors pretty easily:

        P(x) = -2 (x² - 10x + 24)         . . . . common factor of -2

        P(x) = -2 (x - 4) (x - 6)            . . . . leaving an easy quadratic to factor

    The graph of that is a parablola, concave downward because of the negative leading coefficient (-2), with a maximum halfway between the roots at x=4 and x=6.

    The maximum value of P(x) occurs at x=5 (halfway between 4 and 6) and the profit at that point is:

        P(5) = -2(5²) + 20(5) - 48 = -50 + 100 - 48 = 2 thousand dollars.

    If you don't care for graphical reasoning, you could complete the square instead:

        P(x)  = -2(x² - 10x + 24)

                =  -2(x² - 10x + 25 - 25 + 24)   .... add and subtract (-10/2)² = 25

                = -2[(x - 5)² - 25 + 24]              .... factor the first 3 terms as a perfect square

                = -2(x - 5)² + 2                         .... and simplify

    The -2(x - 5)² term is either negative or zero, with maximum of zero, and that happens when x-5 = 0, or x=5.  And the + 2 constant term is the (maximum) profit when the squared term is zero.  That's 2 thousand dollars, just as above.

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  • ted s
    Lv 7
    1 month ago

    P '= 0 when x = 5...thus B)

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