calculus

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 14 in. long and 6 in. wide, find the dimensions of the box that will yield the maximum volume. (Round your answers to two decimal places.)

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  1. The the side of the cut-out square be x inches
    base of box = 14-2x by 6-2x
    (clearly x < 3 )

    volume = V = x(14-2x)(6-2x)
    = 84x - 40x^2 + 4x^3
    dV/dx = 84 - 80x + 12x^2
    = 0 for a max V
    divide by 4
    3x^2 - 20x + 21 = 0
    x = (20 ± √148)/6
    = 5.36 or 1.306 inches, but remember x < 3

    so x = 1.31 correct to 2 decimals

    check: pick a value slightly greater and slightly less than our answer

    let x = 1.3 , V = 1.3(14-2.6)(6-2.6) = 50.388
    let x = 1.31 , V = 1.31(14-2.62)(6-2.62) = 50.38836
    let x = 1.4 , V = 1.4(14-2.8)(6-2.8) = 50.176

    x = 1.31 yields the greatest volume

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  2. The demand function for a certain make of replacement cartridges for a water purifier is given by the following equation where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.

    -0.01x^2-0.3x+19
    Determine the consumers' surplus if the market price is set at $1/cartridge. (Round your answer to two decimal places.)

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  3. Find the point on the graph where the tangent line is horizontal.
    x/(x^2+25)

    I know you have to do the quotient rule to find the derivative of the function, but I do not know what to do after that.

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