M is a rectangular solid. Find the volume of M Statement #1

[BTGmoderatorDC]

M is a rectangular solid. Find the volume of M

Statement #1: The bottom face of M has an area of 28, and the front face, an area of 35.

Statement #2: All three dimensions of M are positive integers greater than one.

OA C

Source: Magoosh

[fskilnik@GMATH]

M is a rectangular solid. Find the volume of M

Statement #1: The bottom face of M has an area of 28, and the front face, an area of 35.

Statement #2: All three dimensions of M are positive integers greater than one.
Source: Magoosh

$$? = abc$$
$$\left( 1 \right)\,\,\left( {ab,ac} \right) = \left( {28,35} \right)\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,28,35} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 28 \cdot 35 \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {2,14,{{35} \over 2}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \cdot 14 \cdot {{35} \over 2} \ne 28 \cdot 35 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,a,b,c\,\, \ge 2\,\,{\rm{ints}}\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {2,2,2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \cdot 2 \cdot 2 \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {2,2,3} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \cdot 2 \cdot 3 \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
ac = 35 \hfill \cr
a,c\,\, \ge 2\,\,{\rm{ints}} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{
\left( {a,c} \right) = \left( {5,7} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{ab\,\, = \,\,28} \,\,\,\,b = {{28} \over 5}\,\,\,\,{\rm{impossible}}\,\,\,\,\left( {b\,\,{\mathop{\rm int}} } \right) \hfill \cr
\,{\rm{or}}\, \hfill \cr
\,\left( {a,c} \right) = \left( {7,5} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{ab\,\, = \,\,28} \,\,\,\,b = 4\,\,\,\, \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 7 \cdot 4 \cdot 5$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Jay@ManhattanReview]

M is a rectangular solid. Find the volume of M

Statement #1: The bottom face of M has an area of 28, and the front face, an area of 35.

Statement #2: All three dimensions of M are positive integers greater than one.

OA C

Source: Magoosh

Given: M is a rectangular solid.

We have to find the volume of M.

We know that the volume of a rectangular solid = a*b*c; where a = = length, b = breadth, and c = height

Let's take each statement one by one.

Statement #1: The bottom face of M has an area of 28, and the front face, an area of 35.

=> ab = 28 and ac = 35.

We can't get the unique values of a, b and c. Insufficient.

Statement #2: All three dimensions of M are positive integers greater than one.

a > 1, b > 1 and c > 1. Cetainly insufficient.

(1) and (2) together

We have

ab = 28 = 2 x 14 = 4 x 7;
ac = 5 x 7

We see that since ac = 5 x 7 (both primes) and in ab, only 7 (prime) is common, a must be 7. Thus, b = 4 and c = 5.

Volume of M = a*b*c = 7*4*5 = A unique value. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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