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find the radius and height of a cylindrical soda can with a volume of 21in^3 that minimize the surface area.

  • math -

    radius -- r
    height --- h
    given: πr^2 h = 21
    h = 21/(πr^2)

    Surface area (SA) = 2πr^2 + 2πrh
    = 2πr^2 + 2πr(21/(πr^2)
    = 2πr^2 + 42/r
    d(SA)/dr = 4πr - 42/r^2
    = 0 for a max/min of SA
    4πr = 42/r^2
    r^3 = 42/(4π)
    r = (42/(4π)^(1/3) = 1.495163.. or appr 1.5
    then h = 21/(πr^2) = 2.99027 or appr 3

    notice the h = 2r

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