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21 April, 10:19

An energy drink container in the shape of a right circular cylinder must have a volume of 19 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches). The cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side. Find the dimensions that will minimize the cost. (Round your answers to two decimal places.)

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  1. 21 April, 13:20
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    the dimension that will minimize cost are radius of 2.79 inch and height of cylinder of 1.40 inch.

    Step-by-step explanation:

    volume of a circular cylinder = πr²h

    volume of 19 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches) = 19 * 1.80469 = 34.28911 in³

    The cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side, means

    surface area of top + bottom = surface area of side

    area of circle (top + bottom) = πr²+πr² = 2πr²

    area of the side = 2πrh

    the cost of top and bottom = twice the cost of side

    2πr² = 2 (2πrh) = 4πrh

    divide both side by 2πr

    r = 4πrh / 2πr = 2h

    r = 2h

    for volume = πr²h = 34.28911 in³

    and r = 2h

    π (2h) ²h = 34.28911 in³

    4πh³ = 34.28911 in³

    h³ = 34.28911 / (4*3.142) = 2.7283 in³

    h = cube root of 2.7283 in³ = 1.397 inch

    r = 2 * 1.76 = 2.794 inch

    the dimension that will minimize cost are radius of 2.79 inch and height of cylinder of 1.40 inch.
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