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What is the volume of the largest cube that can fit

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What is the volume of the largest cube that can fit [#permalink] New post 02 Jun 2018, 06:55
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54% (02:05) correct 45% (00:58) wrong based on 11 sessions
What is the volume of the largest cube that can fit inside a cylinder with radius 2 and height 3?

A) 9
B) 8√2
C) 16
D) 16√2
E) 27
[Reveal] Spoiler: OA

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Re: What is the volume of the largest cube that can fit [#permalink] New post 17 Jun 2018, 17:03
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Hello Brent can you put an ans to this question.

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Re: What is the volume of the largest cube that can fit [#permalink] New post 18 Jun 2018, 06:24
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GreenlightTestPrep wrote:
What is the volume of the largest cube that can fit inside a cylinder with radius 2 and height 3?

A) 9
B) 8√2
C) 16
D) 16√2
E) 27


Let's first inscribe the largest possible square inside the circle
Image


Since the radius of the cylinder is 2, we know that the DIAMETER = 4
Image

Since we have a RIGHT TRIANGLE, we can apply the Pythagorean Theorem to get: x² + x² = 4²
Simplify: 2x² = 16
So, x² = 8, which means x = √8 = 2√2

ASIDE: On test day, you should know the following approximations:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2

So, we get: x = 2√2 ≈ 2(1.4) ≈ 2.8
Image


At this point, we should recognize that, since the height of the cylinder is 3...
Image
...then the LARGEST CUBE will have dimensions 2√2 by 2√2 by 2√2

Volume = (2√2)(2√2)(2√2)
= 8√8
= 8(2√2)
= 16√2

Answer: D

Cheers,
Brent
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Re: What is the volume of the largest cube that can fit [#permalink] New post 20 Jun 2018, 03:20
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GreenlightTestPrep wrote:
GreenlightTestPrep wrote:
What is the volume of the largest cube that can fit inside a cylinder with radius 2 and height 3?

A) 9
B) 8√2
C) 16
D) 16√2
E) 27


Let's first inscribe the largest possible square inside the circle
Image


Since the radius of the cylinder is 2, we know that the DIAMETER = 4
Image

Since we have a RIGHT TRIANGLE, we can apply the Pythagorean Theorem to get: x² + x² = 4²
Simplify: 2x² = 16
So, x² = 8, which means x = √8 = 2√2

ASIDE: On test day, you should know the following approximations:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2

So, we get: x = 2√2 ≈ 2(1.4) ≈ 2.8
Image


At this point, we should recognize that, since the height of the cylinder is 3...
Image
...then the LARGEST CUBE will have dimensions 2√2 by 2√2 by 2√2

Volume = (2√2)(2√2)(2√2)
= 8√8
= 8(2√2)
= 16√2

Answer: D

Cheers,
Brent


Hello Brent, I have a question isn't the volume of a cylinder with r = 2 and height = 3 = \(pi* r^2 * h\) and in this case \(pi * 2^2 * 3\) = \(12pi\) and hence the volume of the cube should be 12pi or closest value less than that and from the option choice E?
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This is my response to the question and may be incorrect. Feel free to rectify any mistakes

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Re: What is the volume of the largest cube that can fit [#permalink] New post 20 Jun 2018, 05:07
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amorphous wrote:

Hello Brent, I have a question isn't the volume of a cylinder with r = 2 and height = 3 = \(pi* r^2 * h\) and in this case \(pi * 2^2 * 3\) = \(12pi\) and hence the volume of the cube should be 12pi or closest value less than that and from the option choice E?


You're right to say that the correct answer choice be LESS THAN 12pi, but it need not be close to 12pi.

For example, we could have a very wide and very short cylinder with radius 1,000 and height 1. The volume of this cylinder would be HUGE (1,000,000pi to be exact).
However, since we need to place a CUBE inside this cylinder, and since all the sides of a cube must be the same length, the largest possible cube that will fit will be a 1x1x1 cube (with a volume of 1)

So, even though 1 is less than 1,000,000pi, it is not close.

Cheers,
Brent
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Re: What is the volume of the largest cube that can fit   [#permalink] 20 Jun 2018, 05:07
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