## Poll

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38 members have voted

Joeman Joined: Feb 21, 2014
• Posts: 2094
April 2nd, 2020 at 2:39:22 PM permalink
Max volume = π

Call h the height of the rectangle and d its width, which is also the cylinder's diameter. The volume of the cylinder is:

V = π(d/2)2 x h

Given the perimeter of the rectangle as 6, the perimeter equation looks like:

2d + 2h = 6

or, solving for h, would be

h = 3 - d

Substitute h into the volume equation:

V = π(d/2)2 x (3 - d)

multiply and you get:

V = 3π/4 x d2 - π/3 x d3

To get a maximum, take the first derivative, dV/dd, and set it equal to 0:

3π/2d - 3π/4d2 = 0

Now, solve for d and you get:

d = 2, which means h = 1.

Plug d and h into the volume equation, and you get:

V = π(2/2)2 x 1

which yields:

V = π
"Dealer has 'rock'... Pay 'paper!'"
ThatDonGuy Joined: Jun 22, 2011
• Posts: 5416
Thanks for this post from: April 2nd, 2020 at 3:14:09 PM permalink
Quote: Joeman

Max volume = π

Call h the height of the rectangle and d its width, which is also the cylinder's diameter. The volume of the cylinder is:

V = π(d/2)2 x h

Given the perimeter of the rectangle as 6, the perimeter equation looks like:

2d + 2h = 6

or, solving for h, would be

h = 3 - d

Substitute h into the volume equation:

V = π(d/2)2 x (3 - d)

multiply and you get:

V = 3π/4 x d2 - π/3 x d3

To get a maximum, take the first derivative, dV/dd, and set it equal to 0:

3π/2d - 3π/4d2 = 0

Now, solve for d and you get:

d = 2, which means h = 1.

Plug d and h into the volume equation, and you get:

V = π(2/2)2 x 1

which yields:

V = π

"V = 3π/4 x d2 - π/3 x d3

To get a maximum, take the first derivative, dV/dd, and set it equal to 0:

3π/2d - 3π/4d2 = 0"

Isn't dV/dd = 3π/2 x d - π x d2?

Also, pardon me for being pedantic, but you didn't show that this makes V a maximum as opposed to a minimum or an inflection point.

charliepatrick Joined: Jun 17, 2011
• Posts: 2600
Thanks for this post from: April 2nd, 2020 at 4:06:27 PM permalink
I agree with above...
Using r saves a lot of 1/4 appearing as follows.

Area of top of cyclinder = Pi r^2.

Vol = h * Pi r^2.
Vol = (3 - 2r) * Pi r^2.
Vol = 3 Pi r^2 - 2 Pi r^3.

dV/dr = 6 Pi r - 6 Pi r^2 = 6 Pi r *(1 - r)
This is 0 when r=0 or r=1.

Clearly when r=0, v=0; and r = 1.5 v=0. Hence at r=1 this is a maximum.

r=1 gets d=2, h=1, Vol = Pi.
Joeman Joined: Feb 21, 2014
• Posts: 2094
April 2nd, 2020 at 5:03:52 PM permalink
Quote: ThatDonGuy

Quote: Joeman

Max volume = π

Call h the height of the rectangle and d its width, which is also the cylinder's diameter. The volume of the cylinder is:

V = π(d/2)2 x h

Given the perimeter of the rectangle as 6, the perimeter equation looks like:

2d + 2h = 6

or, solving for h, would be

h = 3 - d

Substitute h into the volume equation:

V = π(d/2)2 x (3 - d)

multiply and you get:

V = 3π/4 x d2 - π/3 x d3

To get a maximum, take the first derivative, dV/dd, and set it equal to 0:

3π/2d - 3π/4d2 = 0

Now, solve for d and you get:

d = 2, which means h = 1.

Plug d and h into the volume equation, and you get:

V = π(2/2)2 x 1

which yields:

V = π

"V = 3π/4 x d2 - π/3 x d3

To get a maximum, take the first derivative, dV/dd, and set it equal to 0:

3π/2d - 3π/4d2 = 0"

Isn't dV/dd = 3π/2 x d - π x d2?

Also, pardon me for being pedantic, but you didn't show that this makes V a maximum as opposed to a minimum or an inflection point.

You are correct! Not sure how that happened. I did this by hand and then copied it over. Must have lost (or gained!) something in translation!

And yes, I did not demonstrate that it was a maximum. I did check my answer before posting, and the volume decreased using both d = 2.01 and 1.99. I figured that was good enough for an internet message board! :)

d = 0 is also a solution, but obviously not a meaningful one for a maximum volume.
"Dealer has 'rock'... Pay 'paper!'"
Wizard Joined: Oct 14, 2009
• Posts: 24360
April 2nd, 2020 at 6:33:47 PM permalink
I agree that pi is the correct answer. Congratulations to all those who solved it.
It's not whether you win or lose; it's whether or not you had a good bet.
ssho88 Joined: Oct 16, 2011
• Posts: 614
April 2nd, 2020 at 11:07:10 PM permalink
Quote: Wizard

I agree that pi is the correct answer. Congratulations to all those who solved it.

Just use 2 formulas and differentiation( dy/dx) will get it.

h= cylinder height

dV/dr = PI(6r - 6r^2) = 0

r=1

V = PI *(1)^1 *1= PI
Mosca Joined: Dec 14, 2009
• Posts: 3936
April 3rd, 2020 at 6:30:51 AM permalink
Shouldn�t the volume be expressed as cubic units? Am I missing something here? If so, sorry for being not math literate.
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ThatDonGuy Joined: Jun 22, 2011
• Posts: 5416
Thanks for this post from: April 3rd, 2020 at 6:53:09 AM permalink
Quote: Mosca

Shouldn�t the volume be expressed as cubic units? Am I missing something here? If so, sorry for being not math literate.

The units in the volume are cubic units - not the number. If the perimeter is, say, 6 cm, then the maximum volume is π cm3.
Mosca Joined: Dec 14, 2009
• Posts: 3936
April 3rd, 2020 at 7:03:12 AM permalink
Quote: ThatDonGuy

The units in the volume are cubic units - not the number. If the perimeter is, say, 6 cm, then the maximum volume is π cm3.

Thanks, that�s what I thought.
NO KILL I
Ace2 Joined: Oct 2, 2017