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A wrapping paper has dimensions 1 yard by 7 yards. What is the volume

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aeros232
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A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   17 Oct 2012, 19:21
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A wrapping paper has dimensions 1 yard by 7 yards. What is the volume in Ft3 of the largest cube that could be packed using this paper?

(1 yard = 3 Ft)

A. 8
B. 9
C. 18
D. 27
E. 36
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D
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Bunuel
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   17 Oct 2012, 20:59
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aeros232 wrote:
A wrapping paper has dimensions 1 yard by 7 yards. What is the volume in Ft3 of the largest cube that could be packed using this paper?

(1 yard = 3 Ft)

A. 8
B. 9
C. 18
D. 27
E. 36

Show Spoiler
D


Note that it should be mentioned that the side of the largest cube must be an integer (or dimensions should be 1 by 6, not 1 by 7).

The wrapping paper has dimensions \(1*3=3\) feet by \(7*3=21\) feet. Thus, its area is \(3*21=63\) square feet.

Now, the surface area of a cube is \(6a^2\), where \(a\) is the length of a side (a cube has 6 faces and the area of each face is \(a^2\)).

So, it must be true that \(6a^2\leq{63}\).

If \(a=4\), then \(6a^2=96\), which means that we won't have enough paper for such cube.
If \(a=3\), then \(6a^2=54\). The volume of such cube is \(a^3=27\) cubic feet.

Answer: D.

If we won't limit the value of a side to integers only, then: \(6a^2={63}\) --> \(a=\sqrt{10.5}\) --> \(volume=a^3=(\sqrt{10.5})^3\).

Hope it's clear.
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aeros232
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   17 Oct 2012, 21:11
Bunuel wrote:
aeros232 wrote:
A wrapping paper has dimensions 1 yard by 7 yards. What is the volume in Ft3 of the largest cube that could be packed using this paper?

(1 yard = 3 Ft)

A. 8
B. 9
C. 18
D. 27
E. 36

Show Spoiler
D


Note that it should be mentioned that the side of the largest cube must be an integer (or dimensions should be 1 by 6, not 1 by 7).


The wrapping paper has dimensions \(1*3=3\) feet by \(7*3=21\) feet. Thus, its area is \(3*21=63\) square feet.

Now, the surface area of a cube is \(6a^2\), where \(a\) is the length of a side (a cube has 6 faces and the area of each face is \(a^2\)).

So, it must be true that \(6a^2\leq{63}\).

If \(a=4\), then \(6a^2=96\), which means that we won't have enough paper for such cube.
If \(a=3\), then \(6a^2=54\). The volume of such cube is \(a^3=27\) cubic feet.

Answer: D.

If we won't limit the value of a side to integers only, then: \(6a^2={63}\) --> \(a=\sqrt{10.5}\) --> \(volume=a^3=(\sqrt{10.5})^3\).

Hope it's clear.


Great, thanks Bunuel
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   19 Oct 2012, 09:42
Bunuel wrote:
aeros232 wrote:
A wrapping paper has dimensions 1 yard by 7 yards. What is the volume in Ft3 of the largest cube that could be packed using this paper?

(1 yard = 3 Ft)

A. 8
B. 9
C. 18
D. 27
E. 36

Show Spoiler
D


Note that it should be mentioned that the side of the largest cube must be an integer (or dimensions should be 1 by 6, not 1 by 7).

The wrapping paper has dimensions \(1*3=3\) feet by \(7*3=21\) feet. Thus, its area is \(3*21=63\) square feet.

Now, the surface area of a cube is \(6a^2\), where \(a\) is the length of a side (a cube has 6 faces and the area of each face is \(a^2\)).

So, it must be true that \(6a^2\leq{63}\).

If \(a=4\), then \(6a^2=96\), which means that we won't have enough paper for such cube.
If \(a=3\), then \(6a^2=54\). The volume of such cube is \(a^3=27\) cubic feet.

Answer: D.

If we won't limit the value of a side to integers only, then: \(6a^2={63}\) --> \(a=\sqrt{10.5}\) --> \(volume=a^3=(\sqrt{10.5})^3\).

Hope it's clear.


Bunnel

Can you please let me know Why you didn't consider a=2? then the volume would be 8
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   19 Oct 2012, 09:45
Expert Reply
mydreammba wrote:
Bunuel wrote:
aeros232 wrote:
A wrapping paper has dimensions 1 yard by 7 yards. What is the volume in Ft3 of the largest cube that could be packed using this paper?

(1 yard = 3 Ft)

A. 8
B. 9
C. 18
D. 27
E. 36

Show Spoiler
D


Note that it should be mentioned that the side of the largest cube must be an integer (or dimensions should be 1 by 6, not 1 by 7).

The wrapping paper has dimensions \(1*3=3\) feet by \(7*3=21\) feet. Thus, its area is \(3*21=63\) square feet.

Now, the surface area of a cube is \(6a^2\), where \(a\) is the length of a side (a cube has 6 faces and the area of each face is \(a^2\)).

So, it must be true that \(6a^2\leq{63}\).

If \(a=4\), then \(6a^2=96\), which means that we won't have enough paper for such cube.
If \(a=3\), then \(6a^2=54\). The volume of such cube is \(a^3=27\) cubic feet.

Answer: D.

If we won't limit the value of a side to integers only, then: \(6a^2={63}\) --> \(a=\sqrt{10.5}\) --> \(volume=a^3=(\sqrt{10.5})^3\).

Hope it's clear.


Bunnel

Can you please let me know Why you didn't consider a=2? then the volume would be 8


We need the largest cube that could be packed using this paper. Now, the larger the side of a cube larger the volume and since a can be 3, no need to consider smaller lengths.

Hope it's clear.
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   19 Oct 2012, 09:53
Thanks Bunnel i din't read the Q properly and can you please respond the private message on statistics which i have sent you yesterday?

and for the post the link is here

clarification-on-statistics-sd-140907.html
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   19 Oct 2012, 09:54
mydreammba wrote:
Bunuel wrote:
aeros232 wrote:
A wrapping paper has dimensions 1 yard by 7 yards. What is the volume in Ft3 of the largest cube that could be packed using this paper?

(1 yard = 3 Ft)

A. 8
B. 9
C. 18
D. 27
E. 36

Show Spoiler
D


Note that it should be mentioned that the side of the largest cube must be an integer (or dimensions should be 1 by 6, not 1 by 7).

The wrapping paper has dimensions \(1*3=3\) feet by \(7*3=21\) feet. Thus, its area is \(3*21=63\) square feet.

Now, the surface area of a cube is \(6a^2\), where \(a\) is the length of a side (a cube has 6 faces and the area of each face is \(a^2\)).

So, it must be true that \(6a^2\leq{63}\).

If \(a=4\), then \(6a^2=96\), which means that we won't have enough paper for such cube.
If \(a=3\), then \(6a^2=54\). The volume of such cube is \(a^3=27\) cubic feet.

Answer: D.

If we won't limit the value of a side to integers only, then: \(6a^2={63}\) --> \(a=\sqrt{10.5}\) --> \(volume=a^3=(\sqrt{10.5})^3\).

Hope it's clear.




Thanks Bunnel i din't read the Q properly and can you please respond the private message on statistics which i have sent you yesterday?

and for the post the link is here

clarification-on-statistics-sd-140907.html
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   15 Jan 2020, 06:22
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In my opinion the question is wrongly formulated. It asks "What is the volume in Ft3 of the largest cube that could be packed using this paper?". As said in by Bunuel, the largest one is sqrt(10.5)^3. Option D is not the largest cube that could be packed.

Either it should change the information given (say that the sides are integers) or change the question.
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   15 Jan 2020, 14:14
Easier approach to this question:

Wrapping paper yard to foot conversion: 1*3=3 and 7*3=21 so the total space on the wrapping paper is 3 * 21 = 63 foot.
Factorize 63 => 3 * 3 * 7.

\(x^3\) is what we need for a cubed object, the maximum size therefore is the smallest factor of 63 which is 3. This is the biggest number which we can cube within our wrapping paper.

So the answer is \(3^3=27\)
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   05 Feb 2021, 06:27
Hi Experts,

Can anyone tell me if my logic is correct here?

Since we are asked the volume of the largest cube out of the given option, and we know that the volume of the cube is a^3, that means in the answer we need to check which one is the perfect cube and the side could be between 3 and 7, we need to see the the options which is the perfect cube. Out of the options it is 27.

Please help me clear my understanding on this.
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink] New post   09 Jan 2023, 01:23
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Re: A wrapping paper has dimensions 1 yard by 7 yards. What is the volume [#permalink]
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