Problem Solving

What is the ratio of the surface area of a cube to the surface area of a rectangular solid identical to the cube in all ways except that its length has been doubled?


1/4


3/8


1/2


3/5


2

IMO its 1/2....


Length of a cube is x..then lenth of block= 2x

Surface area of cube /surface area of block=(6x^2)/(6*x*2x)

OA - 3/5

Surface are of the cube = 6*a^2 (since there are 6 sides having side a each) - I

Surface area of a rectangular solid = 10*a^2 - II

Hence I/II = 3/5

Let "a" be the side of the cube

Surface area of the cube = 6a^2

The rectangular solid has everything same as cube but length is 2a

Surface area of rectangular solid = 2(LH + HB + LB)

2a * a (L*B)
2a * a (L*H)
a * a (H*B)

2a^2 + 2a^2 + a^2 = 5a^2 * 2 = 10a^2

Ratio of cube to rectangular solid = 6a^2 / 10 a^2 = 6/10= 3/5

sowree for opening a year and a half old post , but how did you get the highlighted part ???

parallel_chase wrote:
2a * a (L*B)
2a * a (L*H)
a * a (H*B)

2a^2 + 2a^2 + a^2 = 5a^2 * 2 = 10a^2

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,

I used plug-in numbers for this one- was 30 sec. ahead of time.

Let's say that the cube has a side=1 (A=6*a^2=6)

If the solid has sides a=1, b=1 and c=2 then A=2*1 + 2*2 + 2*2=10

ratio=6/10=3/5

Whats the formula for surface area for cube and rectangular solid?

ddm wrote:What is the ratio of the surface area of a cube to the surface area of a rectangular solid identical to the cube in all ways except that its length has been doubled?


1/4


3/8


1/2


3/5


2

I will plug in some values .

Surface area of a cube is 6a^2

Let the side of a cube be 1

So surface area of the cube is 6

Now we go to the Cuboid.

Bredth and Height of this cuboid is same as that of the cube = 1

Length of the Cuboid is 2

Now surface area of the cube is

2( l*b + b*h + l*h )

=> 2 ( 2*1 + 1*1 + 2*1 }

=> 2 { 5 } = 10


Now surface area of the cube to the cuboid is 6 /10 = 3/5

anirudhbhalotia wrote:Whats the formula for surface area for cube and rectangular solid?

Surface Area of a Cube = 6 a^2, where "a" is the length of the side of each edge of the cube

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac, where a, b, and c are the lengths of the 3 sides

What is the ratio of the surface area of a cube to the surface area of a rectangular solid identical to the cube in all ways except that its length has been doubled?

surface area of cube=6*a^2
thus ratio = 6*a^2 / (6*(2a)^2) = 1/4
IMO A

surface area of cube is 6a^2 while that of the rectangular solid is 10a^2 so the ratio is 3/5 the answer is D

Plugging in gives me the result 3/4 no matter which way i try it. EG:

3: 2(3^2) + 2(3^2) + 2(3^2) = 6(3^2) = 54

Then, 2(3^2) + 2(3^2) + 2(3*6) = 72

54/72 = 3/4

Where am I going wrong?

disregard question. got it.