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#1 2006-04-16 01:16:26

Stephen
Guest

Logical Rectangle-cylinder deriv

A rectangle has a perimeter of 15cm. What is it`s maximum volume if the rectangle starts spinning around one of it`s sides shaping a cylinder?

I assume one side is: x
I assume one side is: x
The other side is : (15-2x)/2
The other side is : (15-2x)/2

A cylinders volume is : π*r^2*h
Where π=3.14159265359
r: is half of one side as one whole side becomes the cylinders diametre when it starts spinning.
and height is a full opposite site.

I assume it starts spinning on the side (15-2x)/2 which becomes the cylinders diametre and x its hight:
But the forumla for a cylinders volume needs the radius which is: ((15-2x/2)/2) = (15-2x)/4

The formula for the cylinders volume then is: π*x((15-2x)/4)^2

the Above calculation results in a volume for the cylinder as :
(225πx) - (60πx^2)+(4πx^3) /16

As we want to know the maximum volume we derive(typo) it:

V`=37,699x^2 - 376,991x + 706,858
This is a simple 2nd degree equation resulting in x1=(7,5) x2=2,5

Now to get the maximum volume we put 2,5 into the Volume formula since this is a maximum:

(225π*2,5) - (60π*2,5^2) - (4π2,5^3) / 16 = 49,08

BUT HERE is my problem: The answer ought to be 196 and therefor (225π*2,5) - (60π*2,5^2) - (4π2,5^3) / 4
is the correct volume formula for the cylinder.

But since one side of the original rectangle is (15-2x/2) then the raidus for the cylinder is ((15-2x/2)/2)

My fault accurs as I get the forumla for the cylinder to be split by 16, it should be 4, can anyone see where in the transitiation/calculation from rectangle to cylinder and coverting diametre to a radius I do wrong since the above calculation is correct except the volume is to be split by 4 and not as I get 16.

Thanks in advance, Stephan

#2 2006-04-16 01:39:03

ganesh
Moderator
Registered: 2005-06-28
Posts: 13,164

Re: Logical Rectangle-cylinder deriv

Stephen wrote:

I assume it starts spinning on the side (15-2x)/2 which becomes the cylinders diametre and x its hight:
But the forumla for a cylinders volume needs the radius which is: ((15-2x/2)/2) = (15-2x)/4

(15-2x)/2 is not the diameter. It is the radius of the cylinder since the rectangle has one side as its axis and rotates a complete circle (360 degrees).


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#3 2006-04-16 09:43:28

Stephen
Guest

Re: Logical Rectangle-cylinder deriv

OMG, thanks for the answer! Imagine I sat with this for about 1hour yesterday yesterday, I really hope my hangover is eligable as an excuse. So logical, thanks! :>

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