What is the Volume of a Certain Rectangular Solid? GMAT Data Sufficiency

Question: What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
(2) Each of two opposite faces of the solid has area 40.

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Answer: C

Solution and Explanation:

Approach Solution 1:

Because opposite faces in a rectangular solid are equal and all angles are right angles, a rectangular solid can only have a maximum of three distinct areas on each of its faces. On the diagram, this means that the yellow, green, and red faces can all have different areas. I say at most because, for instance, a rectangular solid could be a cube, in which case all of its faces would be equal, but it could also only have two distinct areas of its faces, as in the situation of a square base with a height that is not equal to one of the square's sides.

Volume is defined as Volume=Length*Height*Depth for rectangular solids.

GETTING BACK TO THE PRIMARY QUESTION:

What is a specific rectangular solid volume?

(1) The solid has two neighbouring faces that, on the diagram, are coloured blue and yellow, with blue equal to d and yellow equal to 24, respectively.
Blue is d*h = 15 and yellow is l * h=24.
The two equations, yellow=l*h=24 and V=l*h*d, have a combined total of three unknowns, making it impossible to determine the values of the individual unknowns or their product.
Specifically, if blue=d * h = 15,1, and yellow=l * h=24,1,2, then V=l*h*d=24*1*15 = = 360 and if
Blue=d*h = 5* 3 =15 , yellow = l*h = 8*3 = 24, followed by V=l*h*d = 8*3*5 = 90
Two separate responses are not enough.

(2) Each of the solid's two opposed faces has an area of 40; this just provides the areas of the two opposing faces, which is obviously insufficient.
(1)+(2) From (1): blue=d*h=15, yellow=l*h=24 blue=d*h=15 Yellow= l*h =24 and from (2) each of the solid's two opposing faces has area 40, therefore the red face must be the one that does. red=d∗l=40; therefore, we have three separate linear equations with three unknowns; therefore, we can determine the values of each and calculate V=l*h*d. Sufficient.

Multiplying these three equations will demonstrate how to achieve it: l^2*h^2*d^2 = (l*h*d)^2 = 15* 24* 40, and 24^2 + 5^2 -> V= l * h * d = 24*5 =120.

C is the correct answer.

Approach Solution 2:

1) The slid has areas 15 and 24, which are located on two of its adjacent faces.
2) The solid has an area of 40 on each of its opposing faces.

Let the sides' lengths be a, b, and c. The volume will be abc, and the areas of each side will be ab, bc, and ca.

1. ab=15 ac=24 These prevent us from deriving the volume or the individual values of a, b, or c. Such a MISTAKE
2. bc=40. We cannot determine the volume from this. — NOT ENOUGH

Together, we have:
ab=15 \sac=24 \sbc=40
If you add the LHS and RHS together, you get:
Abc=120 = Volume or (ab)(ac)(bc)=15*24*40 (a2)(b2)(c2)=14400

C is the correct answer.

What is the volume of a certain rectangular solid? - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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