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Calculus

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A metal cylindrical container with an open top is to hold 1 cubic foot. If there is no waste in construction, find the dimensions which will require the least amount of material.

  • Calculus - ,

    V = πr^2 h
    h = 1/(πr^2)

    Surface Area (SA)
    = bottom + collar of cylinder
    = πr^2 + 2πrh
    =πr^2 + 2πr(1/(πr^2)
    = πr^2 + 2/r
    dSA/dr = 2πr - 2/r^2 = 0 for a min of SA
    2πr = 2/r^2
    r^3 = 1/π
    r = 1/π^(1/3) = appr .693 ft
    then h = 1/(π(.693^2)) = .683

  • Calculus - ,

    V = π r ² h = 1ft³
    h = 1/π r ²

    SA = π r ² + 2 π r h
    = π r ² + 2 π r * 1/π r ²
    = π r ² + 2 / r

    SA ' = 2π r - 2 / r² = 0

    2π r = 2 / r²
    r³ = 1/π

    r = 1/ ³√π ft

    h = 1/(π r ²)
    = 1/(π (1/ ³√π) ²)
    = ³√π ² / π
    = 1/ ³√π ft

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