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Dimensions of a rectangular wooden box

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Dimensions of a rectangular wooden box

by umaa » Thu Jul 17, 2008 1:35 am
The inside dimensions of a rectangular 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

ans(b)

Please explain me how is b correct.
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by sudhir3127 » Thu Jul 17, 2008 1:45 am
consider the cuboid...it has 3 different rectangles - 6x8, 8x10 and 6x10 tht form the faces of cuboid.

volume of a cylinder is pi*r^2*h

if the cylinder rests on the 6x8 rectangle, then the max. diameter it can hv is 6 inches and max. height is 10 inches..hence volume = 90pi

consider 8x10 face, max diameter = 8 in. and max vol = 96pi

consider 6x10 face, max diameter = 6 in. and max vol = 72pi

hence, max vol = 96pi when diameter = 8 in.

hence the radius = 4 inches
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by Chitts » Thu Jul 17, 2008 1:51 am
First you have to identify the number of canisters possible. Here with the values you have provided the number would be 3. One canister will be in the base of dimension 6x8 and height 10, the other would be 8x10 with height 6 and last 10x6 with height 8.
Now consider the first can dimensions written above. The maximum radius of the canister can be 3 (6/2). This can be easily know by drawing a rectangle and trying to adjust a circle with the highest possible radius. After you know the radius the volume would be "pie.3(square).10" which is 90pie.
Similarly the other two canisters in question would have volume 96pie (pie.4(square).6) and 72pie (pie.3(square).8).

So maximum volume is 96pie. Hence the radius of that canister is 4.

Hope my explanation is clear for you to understand... :)
Regards,
Chitts
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Re: Dimensions of a rectangular wooden box

by parallel_chase » Thu Jul 17, 2008 1:53 am
umaa wrote:The inside dimensions of a rectangular 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

ans(b)

Please explain me how is b correct.
I also got b as the answer.

When height is same while comparing two cylinders, one which has the larger radius will have the maximum volume.

The canister have to place on one the six faces of the cuboid.

Meaning either radius could be 3 (6/2) or 4 (8/2).

Therefore one with 4 radius will have the maximum volume.
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Re: Dimensions of a rectangular wooden box

by ildude02 » Thu Jul 17, 2008 2:58 pm
parallel_chase wrote:
umaa wrote:The inside dimensions of a rectangular 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

ans(b)

Please explain me how is b correct.
I also got b as the answer.

When height is same while comparing two cylinders, one which has the larger radius will have the maximum volume.

The canister have to place on one the six faces of the cuboid.

Meaning either radius could be 3 (6/2) or 4 (8/2).

Therefore one with 4 radius will have the maximum volume.
Why is no one considering the possibility that 10 can be the diameter or of radius 5 which will then give the max radius when compared to 6 and 8? Can someone clarify this, thanks!
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by ildude02 » Thu Jul 17, 2008 3:01 pm
sudhir3127 wrote:consider the cuboid...it has 3 different rectangles - 6x8, 8x10 and 6x10 tht form the faces of cuboid.

volume of a cylinder is pi*r^2*h

consider 8x10 face, max diameter = 8 in. and max vol = 96pi
Why didn't you consider the possibilty that 10 can be the diameter and 8 the height, which will give the value 25 * 8 = 200pi or the maximum volume?Appreciate your response.
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by agemroy » Fri Jul 18, 2008 8:45 am
The question states "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".

This means that the height of the cylinder is equal to the height of the box.

Hence, we cannot consider the diameter to be 10.
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H x L X W

by olpre4 » Sun Aug 10, 2008 6:14 pm
The problem quotes the dimensions 6" x 8" x 10"?

How are we suppose to know which dimension is height?

Is height quoted at the end or the beginning?

Because in this problem it is quoted in the beginning as 6", but in problem #176 (of the Official GMAT Guide 11th Edition) it is quoted at the end.

I wish GMAT would be consistent in its dimension quoting conventions.

Can someone tell me the best way to approach this type of issue on the GMAT.
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by Senator 153 » Sun Aug 10, 2008 6:30 pm
This question is awesome. Best way to approach it: first, draw one 3-D box to give yourself an idea of what you're looking at and label the edges according to length. Write the formula for the volume of the cylinder: V = Length * PI * radius^2

From the formula, you'll see that increasing the radius is the best way to increase the cylinder's volume.

Trouble is, and this is the catch: you can't have "the biggest radius" of 5 in this particular case because the biggest face is? 10x8, not 10x10. You still want the biggest radius, so you go with 10x8 and your radius is 4. The volume is L*PI*(4)^2. Lenth is six, so volume is 96Pi. You could draw another box where the length of the cylinder is 10, but your radius would drop to three (face is 8x6) and you'd be stuck at 90Pi for the volume.

Comments: agemroy, we can't use a diameter of 10 but not for the reason you've specified. The cylinder literally won't fit in the box. olpre4, it actually doesn't matter which dimension we assign height to.
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Thanks

by olpre4 » Sun Aug 10, 2008 6:59 pm
I see. Thanks!!
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by Martian421 » Mon Sep 08, 2008 5:29 am
awesome explanations..

Thanks everyone for posting this!
~Martin
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please clarify explanation of why 10 can't be used

by ellewoods » Fri Apr 17, 2009 2:47 pm
hi, I'm still struggling to understand the last explanation as to why 10 can't be used as diameter and therefore 5 as radius. please could someone break it down in more detail? thanks.
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I finally get it.

by ellewoods » Sat Apr 18, 2009 7:58 am
Never mind, i finally get it. I hope my explanation makes sense for others who don't understand this: if you tried to use a face that had length/diameter as 10, the corresponding width would also have to be at least 10 as well since we are talking about trying to fit a circle in there that obviously has the same radius throughout. If the width is any less than 10, then a cannister of length/diameter 10 (radius 5) won't be able to fit in there. To get the largest volume, you have to use 8 as the length/diameter and a number at least as big (10) as the width. That leaves 6 for the height. The only way 10 could have worked as length/diameter is if width could also have been 10 but that wasn't given as a possible dimension.
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by Jeff@TargetTestPrep » Tue Dec 12, 2017 4:40 pm
umaa wrote:The inside dimensions of a rectangular 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
In order for the canister to stand upright in the box, the diameter of the canister must fit within the base of the box. Let's test various scenarios to determine which will provide the largest volume of the canister. Remember, the volume of a cylinder = πr^2h. Keep in mind that the height of the cylindrical canister is the same as the height of the box.

Scenario 1:

The base of the box is 6 by 8 and the height is 10. Thus, the diameter of the cylinder = 6, which means the radius = 3.

V = π(3)^2 x 10 = 90π

Scenario 2:

The base of the box is 6 by 10 and the height is 8. Thus, the diameter of the cylinder = 6, which means the radius = 3.

V = π(3)^2 x 8 = 72π

Scenario 3:

The base of the box is 8 by 10 and the height is 6. Thus, the diameter of the cylinder = 8, which means the radius = 4.

V = π(4)^2 x 6 = 96π

Thus, the radius of the cylinder that provides the largest volume is 4.

Answer: B

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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