Calculus

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A solid is formed by adjoining one hemisphere to the top of a cylinder. The total volume of the solid is V (treat as a constant) cubic centimeters. Find the minimum surface area of the solid in terms of V.

  • Calculus -

    with radius r and height h,

    pi r^2 h + 2/3 pi r^3 = v
    so,
    h = 3(v - pi r^2)/(2pi r^3)

    a = pi r^2 + 2pi r h + 2pi r^2
    = 3pi r^2 + 2pi r h
    = 3pi r^2 + 3(v - pi r^2)/r^2
    = 3pi r^2 - 3pi + 3v/r^2

    da/dr = 6pi r - 6v/r^3
    dv/dr = 0 when r^4 = v/pi

    a(∜(v/pi)) = 3pi√(v/pi) - 3pi + 3√(pi*v)
    = 3√pi (2√v - 1)

    as usual, double-check the arithmetic and the algebra. I'm pretty sure about the calculus.

  • Calculus -

    thanks a ton.

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