Poll

7 votes (53.84%)
5 votes (38.46%)
3 votes (23.07%)
2 votes (15.38%)
6 votes (46.15%)
1 vote (7.69%)
2 votes (15.38%)
2 votes (15.38%)
7 votes (53.84%)
5 votes (38.46%)

13 members have voted

Joeman
Joeman
Joined: Feb 21, 2014
  • Threads: 34
  • Posts: 1720
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
ThatDonGuy
Joined: Jun 22, 2011
  • Threads: 92
  • Posts: 4020
Thanks for this post from:
Joeman
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
charliepatrick
Joined: Jun 17, 2011
  • Threads: 30
  • Posts: 1950
Thanks for this post from:
Joeman
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
Joeman
Joined: Feb 21, 2014
  • Threads: 34
  • Posts: 1720
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
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1310
  • Posts: 21483
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
ssho88
Joined: Oct 16, 2011
  • Threads: 41
  • Posts: 422
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.

r = radius of cylinder
h= cylinder height

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

r=1

V = PI *(1)^1 *1= PI
Mosca
Mosca
Joined: Dec 14, 2009
  • Threads: 178
  • Posts: 3785
April 3rd, 2020 at 6:30:51 AM permalink
Shouldnt the volume be expressed as cubic units? Am I missing something here? If so, sorry for being not math literate.
NO KILL I
ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
  • Threads: 92
  • Posts: 4020
Thanks for this post from:
Mosca
April 3rd, 2020 at 6:53:09 AM permalink
Quote: Mosca

Shouldnt 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
Mosca
Joined: Dec 14, 2009
  • Threads: 178
  • Posts: 3785
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, thats what I thought.
NO KILL I
Ace2
Ace2
Joined: Oct 2, 2017
  • Threads: 16
  • Posts: 521
April 3rd, 2020 at 12:59:39 PM permalink
New puzzle:

Imagine the following game, which costs $1.00 to play. At the beginning of each round, a random number between 0 and 1 is generated and then multiplied by $1.00. You get paid a portion of that as follows. If, for instance, youre on round 3 and the random number is 0.8169 then you get paid 0.8169 / (3 * 3.8169) = $0.07.

You can play as many rounds as you want and you keep winnings from all rounds. Total winnings are paid when you decide to stop playing, and that is the only time they are rounded to the nearest cent.

For calculation purposes, you can assume players will play forever. What is the house edge of this game?
Last edited by: Ace2 on Apr 3, 2020
Its all about making that GTA

  • Jump to: