Data Sufficiency

BTGmoderatorDC wrote:K is a rectangular solid. Find the volume of K

(1) a diagonal line across the front face of K has a length of 40
(2) a diagonal line across the bottom face of K has a length of 25

OA E

Source: Magoosh

Say, the length, width and height of K are a, b, and c, respectively.

Thus, volume of K = abc

Let's take each statement one by one.

(1) A diagonal line across the front face of K has a length of 40.

=> a^2 + b^2 = 1600. No information about c. Insufficient.

(2) A diagonal line across the bottom face of K has a length of 25.

Say a is the common side.

=> a^2 + c^2 = 625. No information about c. Insufficient.

(1) and (2) together

We have

a^2 + b^2 = 1600 and
a^2 + c^2 = 625

Two equations and three variable -- can't get the unique values of a, b and c. insufficient.

The correct answer: E

Hope this helps!

-Jay
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BTGmoderatorDC wrote:K is a rectangular solid. Find the volume of K

(1) a diagonal line across the front face of K has a length of 40
(2) a diagonal line across the bottom face of K has a length of 25

OA E

Source: Magoosh

You can easily test values to prove insufficiency.

(1) a diagonal line across the front face of K has a length of 40

This give us limited information about 2 of our dimensions, but nothing about the 3rd. It's easy to imagine 40 as the hypotenuse of a 3 : 4 : 5 triangle, so we can imagine side lengths of 24 and 32. But what's the other dimension? It could be 1, or it could be 100, etc. This doesn't tell us about volume.


(2) a diagonal line across the bottom face of K has a length of 25

We can apply the same logic as with (1). Knowing 1 diagonal gives limited information about 2 of the 3 dimensions, but nothing about the 3rd. Insufficient.

(1) & (2) Together

With statement 1, we could easily imagine nice, clean integers. For this one, we can't quite as easily off the tops of our heads. But... we know that 25^2 = 625, and you might know that 24^2 = 576. The difference is 49, so it turns out that a 7: 24 : 25 is an all-integer right triangle. (Even if you didn't know these exact values, you could picture a triangle with one side length of 24 and a diagonal of 25, so another side length that's quite a bit shorter than 24).

The dimensions of this rectangular solid would be 32*24*7.

We could also imagine a version with a perfectly square bottom face, with side lengths of 25/(sqrt(2)), or approximately 18.


We don't need to calculate the other dimensions - we can see that the more cube-like it is, the greater its volume would be.

By contrast, imagine stretching it out so the depth is negligible, and the bottom length is almost 25. We could have a diagonal of 25 with hardly any depth at all, and a short-but-long front face with a diagonal of 40. Clearly the less regular (the less cube-like) we get, the smaller the volume becomes.


This is insufficient. The answer is E.