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 Post subject: GMAT 3D GeometryPosted: Wed Sep 08, 2010 11:46 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
A similar question is discussed here:
http://800score.com/forum/viewtopic.php?f=3&t=138
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A rectangular box is 12 inches wide, 16 inches long, and 20 inches high. Which of the following is closest to the longest straight-line distance between any two corners of the box?

A. 22 inches
B. 25 inches
C. 28 inches
D. 30 inches
E. 34 inches

(C) We must first determine which two corners are furthest apart in a rectangular box. To determine the corners that are furthest apart in a rectangular box, pick any corner, and think about drawing a line through the center of the box to the furthest other corner. Let's call corners that have this relationship "opposite corners".

The distance D between any two opposite corners in a rectangular solid (whether a perfect cube or not) can be determined very quickly using the Pythagorean Theorem in a 3-dimensional manner:
D² = a² + b² + c², where a, b, and c represent the dimensions of the solid.

Now let's use the formula to determine the distance:
D² = 12² + 16² + 20²
D² = 144 + 256 + 400 = 800.

So D = √800.

Rather than trying to approximate √800, let's square the answer choices and determine which is closest to 800.

We know that 30² = 900, so the correct answer must be either choice (C) or (D). Now let's square the number in choice (C).
28² = 784.

Since 784 is much closer to 800 than 900 is, the correct answer is choice (C).

Another way to solve this question is to use 3:4:5 triangles. Since 12:16:20 is a multiple of 3:4:5, the diagonal is 20. To find the answer, find the hypotenuse of a triangle with that diagonal and the third dimension of the box – 20 and 20, which has a hypotenuse of 20√2. √2 equals approximately 1.4, so the answer is about 28.
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Possible explanation:
√800 = √(4 × 2 × 100) = 20 × √2 (which most people know is 1.41) = 28

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 Post subject: Re: math (test 2, question 23):Posted: Wed Sep 08, 2010 12:11 pm

Joined: Sun May 30, 2010 2:23 am
Posts: 498

Yes, your approximation in this question is possible and works well.

However do not forget about more general method – squaring answer choices. It works better with arbitrary numbers.

EXAMPLE:
Approximating √910 is not that easy even though number "910" looks good.

√910 = √(13 × 7 × 5 × 2)

Approximations of √13 and √7 are not that well-known. And calculating approximation of √910 would be more difficult than using more general method – squaring answer choices.

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