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A cone-shaped paper drinking cup is to be made to hold 30 cm^3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Round your answers to two decimal places.)


Height = ?

Radius = ?

Show work please!!!

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Patrick
1 answer

  1. 1/3 pi r^2 h = 30, so
    h = 90/(pi r^2)

    surface area = 2 pi r s
    where r^2+h^2 = s^2, so

    a = 2 pi r √(r^2 + (90/(pi r^2))^2)
    = 2/r √(pi^2 r^6 + 8100)

    for minimum paper, we need da/dr = 0, so, as wolframalpha so ably shows at

    http://www.wolframalpha.com/input/?i=2+pi+r+%E2%88%9A%28r^2+%2B+%2890%2F%28pi+r^2%29%29^2%29

    r = ∛(45√2/pi)
    so, h = ∛(180/pi)

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    Steve

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