Elementary Analysis

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A company wants to manufacture cylindrical aluminum cans with a value of 1000cm^3 (1 Liter) What radius and height of the can be to minimize the amount of aluminum used? Please help! This is very hard. Thanks.

  • Elementary Analysis -

    You know that

    pi r^2 h = 1000, so
    h = 1000/(pi r^2)

    the surface area is
    a = 2pi r^2 + 2pi r h
    = 2pi r^2 + 2pi r * 1000/(pi r^2)
    = 2pi r^2 + 2000/r

    da/dr = 4pi r - 2000/r^2
    = (4pi r^3 - 2000)/r^2

    Since r > 0, da/dr=0 when

    4pi r^3 = 2000
    r^3 = 2000/4pi = 500/pi
    r = 5 ∛(4/pi) ~= 5.42

    go to wolframalpha.com and enter

    plot y = 2pi r^2 + 2000/r where 3<r<8

    to see where it is minimum

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