Posted by **Anonymous** on Wednesday, June 4, 2008 at 9:24pm.

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 -
**Damon**, Wednesday, June 4, 2008 at 9:56pm
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 -
**drwls**, Wednesday, June 4, 2008 at 9:59pm
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 -
**Anonymous**, Wednesday, June 4, 2008 at 10:39pm
Thanks Damon and drwls!

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