Solving Cylinder Volume from Rectangle Perimeter of 40cm

[danizh]

Question: A rectangle with a perimeter of 40cm is rotated around one of its sides, creating a right cylinder. What is the largest possible volume for this cylinder?
Here's what I have done so far:
Equation #1:
40 = 2h + 2r
r = 20 - h
Equation #2:
Volume = pi*r^2h
= pi(20-h)(h)
= 20pi*h - h^2*(pi)
Derivate of volume: 20pi-2pi*h
10=h
Therefore, r also equals 10.
Thus, the maximum volume is 3141.592cm^3, which is incorrect.
The actual answer is 3723.37cm^3. Any help would be great.

Sorry, but I think this should be in the "Calculus and Beyond" board.
I'm not too sure how to move it there, though.

 

[moose]

Equation #2:
Volume = pi*r^2h
= pi(20-h)(h)
= 20pi*h - h^2*(pi)

I think you forgot to square the radius (20-h) when putting it in...

I may be wrong, I just looked at it quickly and that's what I saw.

EDIT: My way, my calculator now tells me 3723.3691 so yup, that was your mistake

 

[danizh]

Thanks for the help, I really appreciate it.
I'll be more careful next time.

 

[danizh]

I have another question, wouldn't it be more appropriate if the "restraint equation" was 40 = 2h + 4r rather than 40 = 2h + 2r. It just seems to make more sense since if the cylinder is transformed into a rectangle, each side of the triangle would be a diameter (or two times the radius) rather than just the radius, which we are assuming right now. I'm just curious to know why I get the wrong answer if I do it the way that seems to be more logical to me.

 

[HallsofIvy]

I have another question, wouldn't it be more appropriate if the "restraint equation" was 40 = 2h + 4r rather than 40 = 2h + 2r. It just seems to make more sense since if the cylinder is transformed into a rectangle, each side of the triangle[\b] would be a diameter (or two times the radius) rather than just the radius, which we are assuming right now. I'm just curious to know why I get the wrong answer if I do it the way that seems to be more logical to me.

I assume you meant "rectangle" where you wrote "triangle" above. The reason a side of the rectangle is a radius not a diameter is that the rectangle is rotated about one side, not about a center line of the rectangle.

 

[danizh]

Ah, I understand it now! Thanks for clearing that up.
I think the key to the question is that it is rotated to create a right cylinder.

 

[chaosblack]

what did you get for your "r" value? I'm doing somethign wrong...