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655-705 (Hard)|   Geometry|                     
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The inside dimensions of a rectangular wooden box are 6 29 Apr 2020, 17:24
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BANON
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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Normally in these types of problems, the way i think through this is to analyze the formula.

If the formula for my right circular cylinder canister = Pi . R^2 . h

I know, that in order to increase that entire value of the formula the highest leverage can be placed on R since its squared unlike to h (Pi is anyway a constant). Hence my new motive becomes to know how and what will be the maximum radius that can be accommodated in the rectangle box.

Now the rectangle box has three sides 6,8 and 10. If i keep the Length and breadth as (6 and 8) or as (6 and 10) then the maximum diameter will be 6 for the canister (anything bigger than 6 cannot fit - Try to make a diagram if this step is a little difficult to follow)

Hence the best way is to keep the base of the rectangle box with the length and breadth as (8 and 10). Now the maximum diameter can be 8 (Radius = 8/2 = 4)

Hence, B is the answer

Hope this helps a little
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The inside dimensions of a rectangular wooden box are 6 27 Feb 2021, 09:19
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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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Bunuel is there a visual? I still don't understand why the radius can't be 5 if we were to use the 10 x 8 face. Can't you just stretch the cylinder?
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The inside dimensions of a rectangular wooden box are 6 27 Feb 2021, 10:11
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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.

Bunuel is there a visual? I still don't understand why the radius can't be 5 if we were to use the 10 x 8 face. Can't you just stretch the cylinder?
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Can you inscribe a circle with radius of 5 into a 8 by 10 rectangle? It's that simple.
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The inside dimensions of a rectangular wooden box are 6 27 Feb 2021, 10:52
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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.

Bunuel is there a visual? I still don't understand why the radius can't be 5 if we were to use the 10 x 8 face. Can't you just stretch the cylinder?
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Can you inscribe a circle with radius of 5 into a 8 by 10 rectangle? It's that simple.[/quote]

Yeah, I thought about it again. Makes total sense, the problem is I got stuck thinking too linearly. The key here is while the circle with a radius of 5 can be accommodated by 1 side, it cannot be accommodated by the other side.
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The inside dimensions of a rectangular wooden box are 6 21 May 2022, 01:45
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BrentGMATPrepNow
BANON
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 of cylinder = pi(radius²)(height)

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

Answer: B

Cheers,
Brent
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Brilliant post always BrentGMATPrepNow.
How do we decide which number is the base/radius out of 6 x 8 x 10 so we don't end up with wrong calculation of r ?
Read others post here but still not 100% make sense in this aspect. Thanks
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The inside dimensions of a rectangular wooden box are 6 21 May 2022, 07:34
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Kimberly77
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BANON
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 of cylinder = pi(radius²)(height)

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

Answer: B

Cheers,
Brent

Brilliant post always BrentGMATPrepNow.
How do we decide which number is the base/radius out of 6 x 8 x 10 so we don't end up with wrong calculation of r ?
Read others post here but still not 100% make sense in this aspect. Thanks
Show more

This is why we need to test all three possible orientations.
For example, if we place the cylinder's 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)

Conversely, if we place the cylinder's 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)

And so on.

Does that help?
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The inside dimensions of a rectangular wooden box are 6 22 May 2022, 09:12
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Brilliant thanks BrentGMATPrepNow, does it mean r will always be either the smallest or second smallest out of 6 x 8 x 10? Therefore 10 is not the radius in this instance? Thanks
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The inside dimensions of a rectangular wooden box are 6 22 May 2022, 09:28
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Given: 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.

Asked: Of all such canisters that could be used, what is the radius, in inches, of the one that has the maximum volume?

Canister 1: -
Base = 6*8 ; r = 3
Height = 10
Volume = \(\pi r^2 h = \pi 3^2*10 = 90\pi\)

Canister 2: -
Base = 8*10 ; r = 4
Height = 6
Volume = \(\pi r^2 h = \pi 4^2*6 = 96\pi\)

Canister 1: -
Base = 6*10 ; r = 3
Height = 8
Volume = \(\pi r^2 h = \pi 3^2*8 = 72\pi\)

Canister 2 has maximum volume, radius = 4 inches

IMO B
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The inside dimensions of a rectangular wooden box are 6 22 May 2022, 09:29
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Brilliant thanks BrentGMATPrepNow, does it mean r will always be either the smallest or second smallest out of 6 x 8 x 10? Therefore 10 is not the radius in this instance? Thanks
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Let's examine a few scenarios...

If we place the 6 x 8 side on the ground, then the box (and the resulting cylinder) has a height of 10.

In this scenario, the greatest radius of the cylinder's base is 3 (since this would result in a diameter of 6, which would fit in the box)
In other words the biggest cylinder to fit inside the box will have radius 3 and height 10.



If we place the 6 x 10 side on the ground, then the box (and the resulting cylinder) has a height of 8.

In this scenario, the greatest radius of the cylinder's base is 3 (since this would result in a diameter of 6, which would fit in the box)
In other words the biggest cylinder to fit inside the box will have radius 3 and height 8.




If we place the 8 x 10 side on the ground, then the box (and the resulting cylinder) has a height of 6.

In this scenario, the greatest radius of the cylinder's base is 4 (since this would result in a diameter of 8, which would fit in the box)
In other words the biggest cylinder to fit inside the box will have radius 4 and height 6.

Does that help?
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The inside dimensions of a rectangular wooden box are 6 23 May 2022, 05:06
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That's brilliant with visual illustration and thanks BrentGMATPrepNow. Wondering why 10 x 8 is not being tested as side/base in this scenario? Thanks
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The inside dimensions of a rectangular wooden box are 6 23 May 2022, 06:59
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Kimberly77
That's brilliant with visual illustration and thanks BrentGMATPrepNow. Wondering why 10 x 8 is not being tested as side/base in this scenario? Thanks
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All three possible scenarios are tested above.
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The inside dimensions of a rectangular wooden box are 6 23 May 2022, 07:32
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That's brilliant with visual illustration and thanks BrentGMATPrepNow. Wondering why 10 x 8 is not being tested as side/base in this scenario? Thanks

All three possible scenarios are tested above.
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Thanks BrentGMATPrepNow. My apology I mean 10 x 8 x 6 with 10 as r ?
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The inside dimensions of a rectangular wooden box are 6 23 May 2022, 08:18
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Kimberly77
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That's brilliant with visual illustration and thanks BrentGMATPrepNow. Wondering why 10 x 8 is not being tested as side/base in this scenario? Thanks

All three possible scenarios are tested above.

Thanks BrentGMATPrepNow. My apology I mean 10 x 8 x 6 with 10 as r ?
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I think you mean 10 as diameter.
This would never work.
For example if the base is 10 x 8, then a circle with diameter 10, wouldn't fit inside the box,


In other words, if given two dimensions of a rectangle, the diameter of a circle to fit inside the rectangle can't exceed the smallest dimension
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The inside dimensions of a rectangular wooden box are 6 23 May 2022, 09:04
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Brilliant BrentGMATPrepNow, crystal clear now and appreciation for your great helps and insights always are beyond my words :)
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The inside dimensions of a rectangular wooden box are 6 28 Nov 2022, 10:26
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Can anyone explain this question in a easier way and solve it step by step?
Please..
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The inside dimensions of a rectangular wooden box are 6 08 Feb 2023, 06:29
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dipanjandas792
Can anyone explain this question in a easier way and solve it step by step?
Please..
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Hi dipanjandas792,
Thanks for your query.


Please go through the following detailed explanation.


GIVEN:
  • A rectangular box with inside dimensions 6 inches by 8 inches by 10 inches.
  • A cylindrical canister that stands upright when the closed box rests on one of its six faces.


TO FIND:
  • The radius, in inches, of the cylinder that has the maximum volume.


SOLUTION:
First, observe that the rectangular box has three types of base surfaces based on the dimensions. That is,

Type – 1: Two opposite surfaces (base and top), each with dimensions 6 inches by 8 inches.

Type – 2: Two opposite surfaces (base and top), each with dimensions 8 inches by 10 inches.

Type – 3: Two opposite surfaces (base and top), each with dimensions 6 inches by 10 inches.


Now, we must place the canister so that it stands upright when the closed box rests on one of its six faces. That means the circular base of the canister must lie completely inside the box, AND its height must be less than or equal to the height of the box.

Keeping all this in mind, let’s find out which of the above-mentioned three types is possible for forming the base to our canister.


If Type – 1 is taken as the box’s base:
So, the box’s length = 6 inches and breadth = 8 inches, and the height = 10 inches.

Now, the maximum possible DIAMETER of the canister here will be equal to the smaller dimension of the base: 6 inches. Try visualizing this below:
  • Imagine the base alone (without the entire box) as a rectangle of 6 inches × 8 inches, with a circle inscribed in it (the circle is the base of the canister).
  • Now, this circle can have a maximum diameter of 6 inches because if it goes beyond 6 inches, the circle comes out of the rectangle. It cannot be inscribed.
In this case, the maximum possible height of the canister is equivalent to the height of the box, that is, 10 inches.

So, the volume of the canister = π × \((6/2)^2 \)× 10 = 90 × π …(I)


If Type – 2 is taken as the box’s base:
So, the box’s length = 8 inches and breadth = 10 inches, and the height = 6 inches.

Now again, the maximum possible DIAMETER of the canister here will be equal to the smaller dimension of the base: 8 inches. And the maximum possible height of the canister is equivalent to the height of the box, that is, 6 inches.

(Try visualizing this in the same way as done for type-1.)

So, the volume of the canister = π × \((8/2)^2\) × 6 = 96 × π …(II)


If Type – 3 is kept as the box’s base:
So, the box’s length = 6 inches and breadth =10 inches, and the height = 8 inches.

This time, the maximum possible DIAMETER of the canister here will be equal to the smaller dimension of the base: 6 inches. And the maximum possible height of the canister is equivalent to the height of the box, that is, 8 inches.

So, the volume of the canister = π × (6/2)^2 × 8 = 72 × π …(III)


Comparing volumes from (I), (II), and (III), we get the maximum volume in (II). So, the corresponding radius is 8/2 = 4 inches.


And see, we get what is asked!



Hope this helps!


Best,
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The inside dimensions of a rectangular wooden box are 6 03 Jun 2023, 08:23
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To find the radius of the cylindrical canister that maximizes its volume when placed inside the rectangular wooden box, we need to consider the dimensions of the box and the properties of a cylinder.

The inside dimensions of the wooden box are given as 6 inches by 8 inches by 10 inches. Let's assume the height of the canister is h inches, and the radius of the canister is r inches.

For the canister to stand upright when the box rests on one of its faces, the height of the canister should be equal to the shortest dimension of the box, which is 6 inches.

Now, let's consider the maximum volume of the canister that can fit inside the box. The volume of a cylinder is given by V = πr^2h.

Substituting the values, we have:

V = πr^2(6).

To maximize the volume, we want to find the maximum value of r that still allows the canister to fit inside the box.

The largest possible radius for the canister can be half the width or half the length of the box. In this case, half the width is 4 inches.

Therefore, the maximum radius of the canister is 4 inches.

Hence, the answer is (B) 4.
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The inside dimensions of a rectangular wooden box are 6 12 Nov 2025, 00:34
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