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# The inside dimensions of a rectangular wooden box are 6

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Intern
Joined: 20 Feb 2012
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The inside dimensions of a rectangular wooden box are 6  [#permalink]

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23 Feb 2012, 22:20
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55% (hard)

Question Stats:

59% (01:38) correct 41% (01:38) wrong based on 590 sessions

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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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8
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Posts: 52385

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23 Feb 2012, 22:36
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BANON 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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

$$volume_{cylinder}=\pi{r^2}h$$

If the cylinder is placed on 6*8 face then it's maximum radius is 6/2=3 and $$volume==\pi*{3^2}*10=90\pi$$;
If the cylinder is placed on 6*10 face then it's maximum radius is 6/2=3 and $$volume==\pi*{3^2}*8=72\pi$$;
If the cylinder is placed on 8*10 face then it's maximum radius is 8/2=4 and $$volume==\pi*{4^2}*6=96\pi$$;

So, the maximum volume is for $$r=4$$.

Hope it's clear.

P.S. Notice that radii of 5, 6, and 8 (options C, D and E) are not possible.
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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09 Mar 2013, 11:12
3
2
The three different sizes of faces of the rectangular box are:
6 x 8
8 x 10
6 x 10

Now in simple terms the question asks us to find the maximum radius of the cylinder which can be placed inside the box (reference to volume is an unwanted piece of information to solve this question)

The cylinder with the longest radius can logically sit only on the face of the box with the largest area which is 8 x 10.

This face of the box can accommodate a cylinder with diameter equal to the rectangular face's shortest length i.e 8. hence the maximum possible radius for the cylinder will be 4.

Hope the above is clear.
##### General Discussion
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Joined: 22 May 2012
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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16 Jun 2012, 06:37
1
So as a general rule, can we say that the longest side (10 in this case) can never be the Diameter of the circular base? I am having difficulty visualizing the reason of 10 not being a viable option for diameter.
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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16 Jun 2012, 06:48
2
4
planesmypassion wrote:
So as a general rule, can we say that the longest side (10 in this case) can never be the Diameter of the circular base? I am having difficulty visualizing the reason of 10 not being a viable option for diameter.

Ask yourself: on which face of the cube can you place a cylinder with the diameter of 10.

Even if you consider the largest face (8*10), it's one dimension (8) will still be less than the diameter, so you cannot place the cylinder on it.

Hope it's clear.
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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16 Jun 2012, 07:11
Bunuel wrote:
planesmypassion wrote:
So as a general rule, can we say that the longest side (10 in this case) can never be the Diameter of the circular base? I am having difficulty visualizing the reason of 10 not being a viable option for diameter.

Ask yourself: on which face of the cube can you place a cylinder with the diameter of 10.

Even if you consider the largest face (8*10), it's one dimension (8) will still be less than the diameter, so you cannot place the cylinder on it.

Hope it's clear.

Drew a few figures and I think I understand now. When calculating , we can never take the larger side as diameter as the circular base would extend outside of the rectangular face.

And in this case , only 3 cases possible where dia<height. Thanks bro!!
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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07 Sep 2014, 04:53
1
Found this useful tip to be remembered as a thumb rule in another post .

As a rule of thumb, in such problems, select the second largest side as the diameter (note that it is the diameter and one has to calculate the radius by dividing by 2 before calculating the volume). And the left alone smallest side will be the height of the cylinder (as you need the two largest sides to enclose the bottom of the cylinder the only choice left out for height is the smallest side).
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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12 Mar 2015, 07:57
Are there any similar questions to this for practice purposes?

BANON 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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8
Math Expert
Joined: 02 Sep 2009
Posts: 52385
Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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12 Mar 2015, 08:28
2
1
pinkcupcake wrote:
Are there any similar questions to this for practice purposes?

BANON 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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

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a-rectangular-box-has-dimensions-12-10-8-inches-what-is-the-28790.html
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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14 Oct 2015, 04:58
Bunuel wrote:
BANON 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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

$$volume_{cylinder}=\pi{r^2}h$$

If the cylinder is placed on 6*8 face then it's maximum radius is 6/2=3 and $$volume==\pi*{3^2}*10=90\pi$$;
If the cylinder is placed on 6*10 face then it's maximum radius is 6/2=3 and $$volume==\pi*{3^2}*8=72\pi$$;
If the cylinder is placed on 8*10 face then it's maximum radius is 8/2=4 and $$volume==\pi*{4^2}*6=96\pi$$;

So, the maximum volume is for $$r=4$$.

Hope it's clear.

P.S. Notice that radii of 5, 6, and 8 (options C, D and E) are not possible.

what would be the problem if volume= pi* 4^2* 8 ??
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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04 Dec 2015, 20:49
anik19890 wrote:
Bunuel wrote:
BANON 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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

$$volume_{cylinder}=\pi{r^2}h$$

If the cylinder is placed on 6*8 face then it's maximum radius is 6/2=3 and $$volume==\pi*{3^2}*10=90\pi$$;
If the cylinder is placed on 6*10 face then it's maximum radius is 6/2=3 and $$volume==\pi*{3^2}*8=72\pi$$;
If the cylinder is placed on 8*10 face then it's maximum radius is 8/2=4 and $$volume==\pi*{4^2}*6=96\pi$$;

So, the maximum volume is for $$r=4$$.

Hope it's clear.

P.S. Notice that radii of 5, 6, and 8 (options C, D and E) are not possible.

what would be the problem if volume= pi* 4^2* 8 ??

hi anik19890, volume= $$pi* 4^2* 8$$ is not possible because if height = 8, we are only left with dimensions 10 & 6.
Max of a cylinder's diameter from a base with dimensions 10 x 6 is 6, hence radius is $$6/2=3$$.

If the height used is 8, then max volume of the cylinder is $$pi*3^2*8$$ = $$72pi$$
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The inside dimensions of a rectangular wooden box are 6  [#permalink]

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17 Jul 2016, 02:54
BANON 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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

For the base of the crate we have three values to choose from and we need to select any two ==> 3C2=$$\frac{3!}{2!*1!}$$ = 3 ways in which it can be done.
SO we will have three unique scenarios for the dimension of the base; the third value will become the height.
1st Case) Take base of 6 by 8; Radius 6/2=3 and the height 10 ; volume = 9*10 =90
2nd Case) Take base of 6 by 10; Radius 6/2=3 and the height 8; volume = 9*8= 72
3rd Case) Take base of 8 by 10, Radius 8/2=4 and the height 6; volume 16*6=96

The volume is greatest when radius =4
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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06 Mar 2017, 17:03
1
BANON 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 the 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.

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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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16 Mar 2018, 21:10
Is this a valid rule ? "As a rule of thumb, in such problems, select the second largest side as the diameter (note that it is the diameter and one has to calculate the radius by dividing by 2 before calculating the volume). And the left alone smallest side will be the height of the cylinder (as you need the two largest sides to enclose the bottom of the cylinder the only choice left out for height is the smallest side)." Bunuel
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Re: The inside dimensions of a rectangular wooden box are 6  [#permalink]

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19 Apr 2018, 13:57
Top Contributor
BANON 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 the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

There are 3 different ways to position the cylinder (with the base on a different side each time).
You can place the flat BASE of the cylinder on the 6x8 side, on the 6x10 side, or on the 8x10 side

If you place the base on the 6x8 side, then the cylinder will have height 10, and the maximum radius of the cylinder will be 3 (i.e., diameter of 6).
So, the volume of this cylinder will be (pi)(3²)(10), which equals 90(pi)

If you place the base on the 6X10 side, then the cylinder will have height 8, and the maximum radius of the cylinder will be 3 (i.e., diameter of 6).
So, the volume of this cylinder will be (pi)(3²)(8), which equals 72(pi)

If you place the base on the 8x10 side, then the cylinder will have height 6, and the maximum radius of the cylinder will be 4 (i.e., diameter of 8).
So, the volume of this cylinder will be (pi)(4²)(6), which equals 96(pi)

So, the greatest possible volume is 96(pi) and this occurs when the radius is 4

Cheers,
Brent
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Re: The inside dimensions of a rectangular wooden box are 6 &nbs [#permalink] 19 Apr 2018, 13:57
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