Calculus

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Consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of one with maximum volume.

Work:
V=(pi)r^2(h)
h= V/(pi)(r^2)

2(pi)r + h=30

V=(pi)r^2(h)
V'=2(pi)(r)h+2pi(r^2)=0

  • Calculus -

    h + 2 pi r = 30
    so
    h = 30 - 2 pi r
    dh/dr = -2 pi

    pi r^2 h = v
    so yes
    dv/dr = pi r^2 dh/dr + h (2 pi r) dr/dr
    so
    dv/dr = pi r^2 (- 2 pi) + 2 pi r h
    that is 0 for maximum of minimum
    2 pi^2 r^2 =2 pi r h
    pi r =h
    r = h/pi
    then
    h = 30 - 2 pi r = 30 - 2 pi (h/pi)
    h = 30 - 2 h
    3 h = 30
    h = 10
    r = 10/pi

  • Calculus -

    You have to make V a function of r only abd then set the derivative equal to zero.

    V = pi r^2 h

    h = 30 - 2 pi r

    V = pi r^2 * (30 - 2 pi r)
    = 30 pi r^2 - 2 pi^2 r^3

    dV/dr = 60 pi r - 6 pi^2 r^2 = 0
    divide both sides by 6 pi r
    10 - pi r = 0
    r = 10/pi

    Check my math

  • Calculus -

    Thanks Damon and drwls!

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