calculus

7. A cardboard box manufacturer wishes to make open boxes from rectangular pieces of cardboard with dimensions 40 cm by 60 cm by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the cut-out square so that the box has the largest possible volume. Also, find the volume of the box.

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  1. let each side of the cut-outs be x cm
    So the base is 60-2x by 40-2x and the height is x
    Volume = x(60-2x)(40-2x)
    = 4x^3 - 200x^2 + 2400x
    d(volume)/dx = 12x^2 - 400x + 2400
    = 0 for a max of volume
    3x^2 - 100x + 600 = 0 ----- I divided by 4
    x = (100 ± √(2800)/6
    = 25.5 or 7.85

    but clearly 0 < x < 20 or else the sides make no sense.

    so the cut-outs should be 7.85 by 7.85 cm

    Plug 7.85 into my volume expression to get the max volume

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  2. 6. A right circular cylinder is to be designed to hold 750 cm3 of processed milk, and to use a minimal amount of material in its construction. Find the dimensions for the container.

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  3. A right circular cylinder is to be designed to hold 750 cm3 of processed milk, and to use a minimal amount of material in its construction. Find the dimensions for the container.

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  4. If you had a device that could record the Earth's population continuously, would you expect the graph of population versus time to be a continuous ( unbroken) curve? Explain what might cause breaks in the curve.

    I'm confused. Thanks in advance for any help.

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  5. Tony has a rectangular piece of cardboard which is 60 cm by 40 cm. By cutting squares from each corner of the cardboard and turning up the sides, can he form an open-topped box which will hold (a) 6656 cubic centimeter? (b) 9000 cubic centimeters? (c) what is the maximum amount of space he can obtain?

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