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Anonymous asked in Science & MathematicsMathematics · 1 month ago

Highschool Math HELP!! (Polynomial Functions)?

Question: The volume of a cylindrical can is V(x) = 2πx^3 - 20πx^2 + 64πx - 64π. Determine the possible linear expression for the radius and the height of the cylinder algebraically.

I really don't know how to solve this question. And I've tried numerous of methods but it's not getting me anywhere. If anyone can help me out with this question, I'd appreciate it! Thanks.

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    2pi * x^3 - 20pi * x^2 + 64pix - 64pi

    2pi * (x^3 - 10x^2 + 32x - 32)

    x^3 - 10x^2 + 32x - 32

    Possible rational roots: -32 , -16 , -8 , -4 , -2 , -1 , 1 , 2 , 4 , 8 , 16 , 32

    x^2 * (x - 10) + 32 * (x - 1)

    So, let's try some smaller values for x

    x = 4

    4^2 * (4 - 10) + 32 * (4 - 1) =>

    16 * (-6) + 32 * 3 =>

    -96 + 96 =>

    0

    (x - 4) * (ax^2 + bx + c) = x^3 - 10x^2 + 32x - 32

    ax^3 + bx^2 + cx - 4ax^2 - 4bx - 4c = x^3 - 10x^2 + 32x - 32

    ax^3 = x^3

    a = 1

    bx^2 - 4ax^2 = -10x^2

    b - 4a = -10

    b = 4a - 10

    b = 4 - 10

    b = -6

    cx - 4bx = 32x

    c - 4b = 32

    c = 4b + 32

    c = 4 * (-6) + 32

    c = -24 + 32

    c = 8

    (x - 4) * (x^2 - 6x + 8) = x^3 - 10x^2 + 32x - 32

    (x - 4) * (x - 4) * (x - 2) = x^3 - 10x^2 + 32x - 32

    2pi * (x - 4)^2 * (x - 2)

    Volume of a cylinder is pi * r^2 * h

    r = x - 4

    h = 2 * (x - 2)

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