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The inside dimensions of a retangular wooden box are 6 [#permalink]

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22 Feb 2008, 04:07

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191. The inside dimensions of a retangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical canister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such canisters that could be used, what is the radius, in inches, of the one that has maximum volume?

A. 3 B.4 C. 5 D. 6. E. 8

Hi friends!, please can help me a shortcut for this? many thanks
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191. The inside dimensions of a retangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical canister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such canisters that could be used, what is the radius, in inches, of the one that has maximum volume?

A. 3 B.4 C. 5 D. 6. E. 8

Hi friends!, please can help me a shortcut for this? many thanks

B is the answer. Here is an easy way to solve it. there are 3 options: the high of the canister is equal to 10,8 or 6.

if it's equal to 10 -> the radius is min(8,6)/2 = 3 -> volume is 10*pi*3^2 = 90*pi if it's 8 -> radius is min(10,6)/2 = 3 -> volume obviously below the first one if it's 6 -> radius is min(10,8)/2 = 4 -> volume is 6*pi*4^2 = 96*pi -> radius is 4 -> B

Diameter will never be greater than 8 because it should be equal to the "shorter" side, so only A or B can be possible answers. Since volume is proportional to d^2*h, just do the math for these options.

191. The inside dimensions of a retangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical canister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such canisters that could be used, what is the radius, in inches, of the one that has maximum volume?

A. 3 B.4 C. 5 D. 6. E. 8

Hi friends!, please can help me a shortcut for this? many thanks

Alright, here is my simple and effective way to approach this problem. A cylinder has a base and height. The base is obviously going to rest on one of the many faces of the rectangular box. Lets say it rests on the face with dimensions (6,8) in which case the height of the cylinder will be 10. Remember the face on which the cylinder rests, the radius of the cylinder will be half of the length of the smaller edge of the face. So if the cylinder rests on face (6,8), the radius is going to be 3 inches. Note it cannot be 4 inches because if it is then it will go beyond the boundaries of the (6,8) plane. If you don't agree with this, take a Campbell's soup can, keep it on a rectangular paper and go stretch your imagination. So now we the following options to place the cylinder 1) Base on (6,8). Radius = 3, height = 10. Volume = Pi*R^2*H = 90Pi. 2) Base on (6,10). Radius = 3, height = 8. Dont bother calculating. It is going to be less than 1. 3) Base on (10,8). Radius = 4, height = 6. Volume = 96Pi. Other combinations will either yield a lower volume or be same as 1 and 3. So radius of 4 will bring about max volume for the cylinder.

Can someone elaborate on this explanation...I dont see how the cyclinder could have a radius greater than 3 if it is to fit into the box

the narrowest point of the box is 6; how could something with a radius of 4 (diameter of 8) fit in there

Same here.

If 6 becomes the height of the box, then the two remaining side 10, 8 will be l and b and are the only ones that can limit the radius of the cylinder. The minimum of the two which is 8 will become the diameter.

191. The inside dimensions of a retangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical canister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such canisters that could be used, what is the radius, in inches, of the one that has maximum volume?

A. 3 B.4 C. 5 D. 6. E. 8

Hi friends!, please can help me a shortcut for this? many thanks

B is the answer. Here is an easy way to solve it. there are 3 options: the high of the canister is equal to 10,8 or 6.

if it's equal to 10 -> the radius is min(8,6)/2 = 3 -> volume is 10*pi*3^2 = 90*pi if it's 8 -> radius is min(10,6)/2 = 3 -> volume obviously below the first one if it's 6 -> radius is min(10,8)/2 = 4 -> volume is 6*pi*4^2 = 96*pi -> radius is 4 -> B

Had jumped to (C) without figuring out the implications. Nicely explained. +1 to poster and maratikus' solution
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I will do it this way , in order to get maximum volume of cylinder that can fit in the rectangular box either we maximise the radius or maximise the height

Ist case : maximum height =10 face on that cylinder rest 6X8 , So maximum radius can be 3 volume = pi x 9 x 10=90pi

ii case : maximum radius= can only be 4 as radius of 5 wont fit in as other side is maximum 8 so 10x8 gone to accomodate radius and left 6 for height volume = pi x 16 x 6 = 96pi

I will do it this way , in order to get maximum volume of cylinder that can fit in the rectangular box either we maximise the radius or maximise the height

Neeraj.kausha1 !, your logic is very nice! I love it. Thank you!
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