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Diagonals of a Cube

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Diagonals of a Cube

by edvhou812 » Wed May 11, 2011 7:19 pm
If the box pictured to the right is a cube, then the difference in length between line segment BC and line segment AB is approximately what fraction of the distance from A to C?

Image

A - 10%
B - 20%
C - 30%
D - 40%
E - 50%

Answer: C

This question has been bugging me. The method I used had the following:

AC=x <Side of Cube>
AB=x*sqrt(2) <40-40-90 triangle for the diagonal of a face>
BC=sqrt(3x) <formula for space diagonal of a cube>

However I have gotten the question wrong numerous times with this method because I get the wrong value for BC. Official explanation says I should use the Pythagorean Theorem to determine the value of BC, but I'm perplexed as to why my method did not work. Is anyone able to give some insight?
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by Anurag@Gurome » Wed May 11, 2011 8:47 pm
edvhou812 wrote:If the box pictured to the right is a cube, then the difference in length between line segment BC and line segment AB is approximately what fraction of the distance from A to C?

Image

A - 10%
B - 20%
C - 30%
D - 40%
E - 50%

Answer: C

This question has been bugging me. The method I used had the following:

AC=x <Side of Cube>
AB=x*sqrt(2) <40-40-90 triangle for the diagonal of a face>
BC=sqrt(3x) <formula for space diagonal of a cube>

However I have gotten the question wrong numerous times with this method because I get the wrong value for BC. Official explanation says I should use the Pythagorean Theorem to determine the value of BC, but I'm perplexed as to why my method did not work. Is anyone able to give some insight?
Solution:
Let AC be of length x.
The line segment BC is sqrt(x^2 + x^2 + x^2) = x*sqrt3.
Line segment AB is sqrt(x^2 + x^2) = x*sqrt2.
Difference in length of BC and AB is x*(sqrt3 - sqrt2) = x*(1.732 - 1.414) = x*(0.3)
So, required % is 0.3x/x * 100 = 30%.
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by edvhou812 » Wed May 11, 2011 9:00 pm
Anurag@Gurome wrote:
edvhou812 wrote:If the box pictured to the right is a cube, then the difference in length between line segment BC and line segment AB is approximately what fraction of the distance from A to C?

Image

A - 10%
B - 20%
C - 30%
D - 40%
E - 50%

Answer: C

This question has been bugging me. The method I used had the following:

AC=x <Side of Cube>
AB=x*sqrt(2) <40-40-90 triangle for the diagonal of a face>
BC=sqrt(3x) <formula for space diagonal of a cube>

However I have gotten the question wrong numerous times with this method because I get the wrong value for BC. Official explanation says I should use the Pythagorean Theorem to determine the value of BC, but I'm perplexed as to why my method did not work. Is anyone able to give some insight?
Solution:
Let AC be of length x.
The line segment BC is sqrt(x^2 + x^2 + x^2) = x*sqrt3.
Line segment AB is sqrt(x^2 + x^2) = x*sqrt2.
Difference in length of BC and AB is x*(sqrt3 - sqrt2) = x*(1.732 - 1.414) = x*(0.3)
So, required % is 0.3x/x * 100 = 30%.
I see. I goofed the space diagonal formula. Thanks for the response.
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by GMATGuruNY » Thu May 12, 2011 3:49 am
edvhou812 wrote:If the box pictured to the right is a cube, then the difference in length between line segment BC and line segment AB is approximately what fraction of the distance from A to C?

Image

A - 10%
B - 20%
C - 30%
D - 40%
E - 50%

Answer: C

This question has been bugging me. The method I used had the following:

AC=x <Side of Cube>
AB=x*sqrt(2) <40-40-90 triangle for the diagonal of a face>
BC=sqrt(3x) <formula for space diagonal of a cube>

However I have gotten the question wrong numerous times with this method because I get the wrong value for BC. Official explanation says I should use the Pythagorean Theorem to determine the value of BC, but I'm perplexed as to why my method did not work. Is anyone able to give some insight?
Let s = side of each square face.

It is helpful to have memorized the following:
Diagonal of the cube = √(3s^2).
Diagonal of each square face = s√2.

Let AC = 1.
BC = √(3* 1^2) = √3.
AB = 1*√2 = √2.
BC - AB = √3-√2 ≈ 1.7 - 1.4 ≈ .3.
Difference/AC = .3/1 = 30/100 = 30%.

The correct answer is C.
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by djiddish98 » Thu May 12, 2011 7:48 am
I've seen the BeatTheGMAT flash cards indicate that we should know the actual value of sqrt(3) and sqrt(2), but I'd never seen a question that actually required that knowledge.

So should I know that sqrt(3) = 1.7 and sqrt(2) = 1.4 - or is this an odd question?
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by GMATGuruNY » Thu May 12, 2011 8:12 am
djiddish98 wrote:I've seen the BeatTheGMAT flash cards indicate that we should know the actual value of sqrt(3) and sqrt(2), but I'd never seen a question that actually required that knowledge.

So should I know that sqrt(3) = 1.7 and sqrt(2) = 1.4 - or is this an odd question?
Knowing that √3 ≈ 1.7 and that √2 ≈ 1.4 can certainly be helpful.
Consider the following problem:

Image

One approach would be to plug in a value for n.
Let n = 2.
Then 1/(√(n+1) - √n)
= 1/(√3-√2)
≈ 1/(1.7-1.4)
= 1/.3
= 10/3.

Now we plug n=2 into all the answers to see which yields a value close to 10/3.

Only answer choice E works:
√(n+1) + √n
= √(2+1) + √2
= √3 + √2
≈ 1.7 + 1.4
≈ 3.1, which is very close to 10/3.

The correct answer is E.
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by djiddish98 » Thu May 12, 2011 9:36 am
Thanks for the example - it makes sense to come equipped with as many tools in the tool belt as possible, I suppose!
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