Diagonals of a Cube

Question

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?

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?

Anurag@Gurome · Answer

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?

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%.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

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edvhou812 · Answer

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? 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 AB=x*sqrt(2) <40-40-90 triangle for the diagonal of a face> BC=sqrt(3x) 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.

GMATGuruNY · Answer

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?

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%.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/

djiddish98 · Answer

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?

GMATGuruNY · Answer

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:

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.

djiddish98 · Answer

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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