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Geometry problem

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by Troika » Thu May 17, 2012 7:36 am
A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the largest possible volume of a right cylinder that is placed inside the box?

OA: 200pie

Source: Manhattan GMAT Geometry 4th edition, pg. 49.
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by LalaB » Thu May 17, 2012 9:51 am
the largest volume is possible, if the height is 8, and the rest (10 and 12) are the legs.

so, the volume =5^2*8=25*8=200
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by Thrills4ever » Thu May 17, 2012 6:34 pm
I think it should be 200pi since V = pi*r^2*h
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by Anurag@Gurome » Thu May 17, 2012 8:00 pm
HG10 wrote:A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the largest possible volume of a right cylinder that is placed inside the box?

OA: 200pie

Source: Manhattan GMAT Geometry 4th edition, pg. 49.
Volume of cylinder = pi * r² * h.
For maximum volume, radius should be maximum because it has a square power.
This is possible if we take the base of cylinder as the base of the box which is 12*10.
Then the diameter is 10 for the cylinder, as it has to be lesser of 10 and 12.
Or radius is 10/2 = 5 and height is 8.
So volume is pi * 5² * 8 = 200pi.
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by Troika » Sat May 19, 2012 9:55 pm
Guys, thank you for your responses.

I just have one question, i.e. if radius should me maximum for maximum volume, why select r=10/2, why not r=12/2 and h=10 instead of h=8?
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by LalaB » Sat May 19, 2012 11:42 pm
HG10 wrote:Guys, thank you for your responses.

I just have one question, i.e. if radius should me maximum for maximum volume, why select r=10/2, why not r=12/2 and h=10 instead of h=8?
u better try to draw ur desired figure. if u do so, u will find out that it is impossible to have two sides 10 and 12, and get 12/2,since the max value of ur side is 10.
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by darontan » Sun May 20, 2012 12:01 am
thanks
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by Troika » Sat May 26, 2012 7:00 am
Thank you, LalaB and Anurag@Gurome for your solutions and explanations.

@LalaB, thanks for pointing out the dimensional limitations.
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