Seven identical cubes are put together in a line to form a cuboid. Wha

Question

Seven identical cubes are put together in a line to form a cuboid. What is the approximate percentage loss in total surface area?

(a) 20%
(b) 30%
(c) 40%
(d) 50%
(e) more than 50%

ScottTargetTestPrep · Accepted Answer

Solution:

Individually, each cube has 6 faces. Therefore, 7 cubes have a total of 6 x 7 = 42 faces and the total area of the 42 faces is the total surface area of the 7 cubes.

Now, when the 7 cubes are put together in a line, the 2 cubes at either end of the line each lose 1 face, while the 5 cubes in between each lose 2 faces because they are now hidden from view. Therefore, a total of 2 x 1 + 5 x 2 = 12 faces are hidden from view. Since 12/42 = 2/7 ≈ 0.3, approximately 30% of the total surface area is lost.

Alternate Solution:
 
Let x be the side length of the cubes. Then, the surface area of one cube is 6x^2 and the surface area of 7 cubes is 7 * 6x^2 = 42x^2.

When the 7 cubes are put together in a line, we obtain a cuboid with dimensions 7x, x and x. The surface area of this cuboid is 2(7x^2 + 7x^2 + x^2) = 30x^2.

Thus, the approximate percentage loss in total surface area is (42x^2 - 30x^2)/42x^2 = 12/42 ≈ 0.3.

Answer: B

Dashingnightmare · Answer

Let the sides of cube is a
TSA of 7 cubes= 7*6a^2 =42a^2

For cuboid : Leanth= 7a, breadth =a, height=a
TSA = 2(7a^2+a^2+7a^2)=30a^2

Percentage loss in Surface area = (42a^2-30a^2)/42a^2 =3/7 = approx 30% , Hence IMO B is the correct answer

GMATWhizTeam · Answer

A cube has six square faces of area  a^2  each, where 'a' is the length of the edge of the cube. 
Therefore, total surface area of cube = 6 *  a^2

Total surface area of 7 cubes = 6 *  a^2  * 7 = 42 *  a^2 .

When 7 cubes are joined together in a line, two side faces of each cube do not count towards the surface area of the cuboid, wherever two cubes are joined.

When 7 cubes are joined, 6 joints are created. At each joint, we do not count two faces of area  a^2  each.

Surface area of faces not counted = 6 * 2 *  a^2  = 12 *  a^2

Percentage loss in surface area =  \frac{12 * a^2 }{ 42 * a^2 } * 100 =  \frac{4 }{ 14}  * 100 = 28.56%

The closest percentage to 28.56% is 30%.

The correct answer option is B.

Abhishek009 · Answer

Let, each face be 10 units...

TSA of the cubes will be  7*6*10^2 = 4200

After placing cubes side by side,

Length will be  7*10  ; Breadth will be  10  & Height will be  10

So, TSA of Cuboid will be  2(70*10 + 70*10 + 10*10) = 3000

Thus, loss in area will be  \frac{(4200 - 3000)}{4200}*100 =  28.xx ~  30 % Answer must be (B)

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Seven identical cubes are put together in a line to form a cuboid. Wha

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