Derivatives Application NEED HELP QUICK PLZZ!?

A rectangular open-topped box is to be made from a rectangular piece of material by cutting a square from each corner and folding up the sides. The rectangular piece of material is 30 cm long and the base of the box will be three times as wide as it is high. Find the size of the square to be removed in order to maximize the volume of the box. State the maximum volume.

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  • 1 year ago
    Favorite Answer

    .

    box dimensions:

    height = x

    width = 3x

    length = 30 - 2x

    V = x ( 3x ) ( 30 - 2x )

    V = 90x² - 6x³

    V’(x) = 180x - 18x²

    180x - 18x² = 0

    18x( 10 - x ) = 0

    x = 0, x = 10

    V’(x) = 180x - 18x²

    V’'(x) = 180 - 36x

    V’’( 0 ) = 180, tests for minimum

    V’’(10) = -180, tests for maximum

    ∴ side of square = 10 cm

    ━━━━━━━━━━━

    maximum volume:

    V = 90x² - 6x³

    V = 90*10² - 6*10³

    V = 3000 cm³

    ━━━━━━

  • Anonymous
    1 year ago

    This is the most standard calculus problem ever. Your teacher probably did this exact same problem in class with different numbers. And I guarantee it's in your textbook.

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