# Related rates maxium voulme of cylinder

1. Apr 18, 2012

### taco

If 1200∏ cm^2 of material is available to make a cylindrical can with a circular base an open top, find the largest possible volume of the can.

the formulas i used:
v=∏* r^2 * h
surface area = 2∏r^2 + 2rh

my attempt:
1200=r^2h∏
h=1200/r^2∏

SA=2∏r^2+2∏r(1200/∏r^2)
=2∏r^2+ 2(1200/r)
S`(A)=4∏r-1200/r^2

and i don't know where to go from here, assuming i did this correctly

2. Apr 18, 2012

### LCKurtz

"Open top" means no top. You have included both a top and bottom. Also your lateral area is wrong. It would be circumference times height.

You didn't. Why did you set the volume equal to $1200$?

3. Apr 18, 2012

### taco

4. Apr 18, 2012

### LCKurtz

It gave you $1200\pi$ sq. cm. of metal, not $1200$. And $cm^2$ is an area, not a volume. So start by setting the [correct] surface area equal to $1200\pi$ and maximize the volume, which is what the problem asks for, after all.

5. Apr 18, 2012

### taco

So the correct formula would be
1200pi=2*pi*r^2+pi*r*h
but because it is an open top it would be
1200pi=pi*r^2+pi*r*h
and were trying to maximize the volume, v=pi*r^2*h
h=1200pi/(pi*r^2+pi*r)

v=pi*r^2*((1200pi/(pi*r^2+pi*r))
am i on the right track now?

6. Apr 18, 2012

### LCKurtz

Yes, but you still don't have the correct formula for the lateral area (circumference times height) and your algebra solving for h to substitute is bad. Once you fix those you will have V as a function of r to maximize using usual calculus methods.

7. Apr 18, 2012

### sharks

The total area of an open top cylinder = area of the bottom circle + area of the curved side.
Your error lies with finding the second part (the area of the curved section).
The formula for the volume is correct.

Using substitution to solve for V, your answer should be in terms of r.

8. Apr 18, 2012

### taco

oh okay, thank you

9. Apr 18, 2012

### sharks

The answer should be: $8000\pi\,cm^3$

10. Apr 18, 2012

### taco

sharks is it possible if you show me your steps?

11. Apr 18, 2012

### sharks

You know, i hate it when this happens.:rofl: I can't do your homework, but i can i try to guide you closer to the solution. This should simplify what LCKurtz already told you:

First, you deduce the correct expressions for volume and total area. Those are fundamental steps. You need to equate the formula for the area with the value given in your problem itself. Then you need to use substitution into the volume, as you're trying to find the volume. That should be simple enough.

Now, here is the interesting part... What do you do when you need to find the maximum value of a variable which is dependent upon another variable? (i hope that i am phrasing this idea correctly here). Imagine a graph of V against the expression in terms of r, where the graph peaks at some point to give you the maximum volume. How would you find the maximum value of V based on the expression in terms of r? In other words, you need to find the value of r which corresponds to the maximum value of V.

12. Apr 18, 2012

### taco

alright thanks sharks and lckurtz, ill try my best

13. Apr 18, 2012

### sharks

Hint: Find the stationary point, hence find the stationary value.

14. Apr 18, 2012

### Staff: Mentor

Homework Help:
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On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given. Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made.

15. Apr 18, 2012

### taco

sorry about that mark, just that im very confused at this

16. Apr 19, 2012

### sharks

For a graph of V against r, how do you find the maximum point?