Hi Bhumika,
Jumping in here, but since it's an old post, I thought I might try to clarify what the expression you've highlighted represents.
For a cube with sides of length A, the rectangular solid described will have the following dimensions:
Length: 2A
Base: A
Height: A
Of course, we have:
2 sides defined by Length x Height
2 sides defined by Height x Base
2 sides defined by Base x Length
The surface area formula used by the original poster, for a rectangular solid, thus represents double the sum of three adjacent sides. In other words, for the SA of any rectangular solid, you must find the area of each of three unique sides (represented by LB, LH and BH), add them up, and double the whole mess.
In this case, LH and LB have areas of 2A^2, and BH has an area of A^2. So the sum of three adjacent/unique sides is 5A^2, and then that is doubled to determine the total surface area. Hope that clarifies the expressions you've highlighted!
From a strategy standpoint, I'd also like to suggest two methods by which this problem can probably be answered quickly and without the introduction of variables.
The first is to plug in a value:
If the cube has a side of length 1, its surface area is 6 square inches. The surface area of the rectangular solid would be the sum of four sides with areas of 2 square inches and two sides with areas of 1 square inch. Its total surface area would be 10 square inches, for a ratio of 6/10 which simplifies to 3/5.
The other way is simply an intuitive approach:
Imagine two cubes stacked next to each other. One cube alone would have six sides exposed, whereas two cubes stacked together represent 12 sides, only 10 of which are exposed. So the number of cube-sized sides exposed (that's a mouthful) are six for a cube and 10 for the rectangle. Thus, the ratio is 3/5.
Hope one or all of those approaches helps!
Cheers,
Stephen
GMAT Instructor
Knewton Inc.
