If a cube with the length of the side of 4 cm is cut into smaller cube

Question

If a cube with the length of the side of 4 cm is cut into smaller cubes with the length of the side of 1 cm, then what is the percentage increase in the surface area of the resulting cubes?

A. 4%
B. 166%
C. 266%
D. 300%
E. 400%

M01-11

Bunuel · Accepted Answer

A cube has 6 faces.
 
The surface area of a cube with the length of the side of 4 cm is  6*4^2=6*16   cm^2 .
 
Now, since the volume of the big cube is  4^3=64   cm^3  and the volume of the smaller cubes is  1^3=1   cm^3 , then when the big cube is cut into the smaller cubes we'll get  \frac{64}{1}=64  little cubes. Each of those little cubes will have the surface area equal to  6*1^2=6   cm^2 , so total surface are of those 64 little cubes will be  6*64   cm^2 .
 
 6*64  is 4 times more than  6*16  which corresponds to 300% increase.
 
 Or:  general formula for percent increase or decrease, (percent change):  	ext{Percent} = \frac{	ext{Change}}{	ext{Original}}*100 
 
So the percent increase will be:  Percent=\frac{	ext{Change}}{	ext{Original}}*100=\frac{6*64-6*16}{6*16}*100=300% .

Answer: D

rohu27 · Answer

Hi,

Can you please elaborate this:

when the cube is cut into the smaller cubes with side of 1cm we'll get 4*4*4=64

Thanks

Bunuel · Answer

The big cube which dimensions 4*4*4 can "produce" 4*4*4 small cubes with dimensions of 1*1*1 (base layer can give 4*4 small cubes and as there are 4 layers than total of 4*4*4 small cubes).

pratsh123 · Answer

Total Surface Area of bigger cube = 6a^2 = 6*(4)^2 = 96
Total Surface Area of 1 smaller cube = 6*a^2 = 6*(1)^2 = 6

Let  'x ' be total number of small cubes formed after cutting
Volume of bigger cube = (x) Volume of smaller cube => 4^3 = (x)1^3 =>  x = 64 
Hence,total surface area of 64 smaller cubes = 64 * 6 = 384

Now % increase in Total Surface Area =  *100=>  *100 =  300%  
Hence  Ans (D)

PareshGmat · Answer

Big cube surface area = 4*4*6 = 96

Small cubes surface area (Total) = 6 *64

% = 6*64*96/100 = 400

% Increase = 400-100 = 300% = Answer = D

Bunuel · Answer

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amirsalehi · Answer

I have problem with its verbal... 4cm cube! Does it mean that each side is 4cm or the area is 4cm...? is it popular to use this language in Geometry?

Bunuel · Answer

The area cannot be in units, it's in square units, so 4cm can only mean length of something.

davidfrank · Answer

Hey Bunuel,

My question might sound stupid but why are you going back and calculating the volume of the bigger cube. Why can't we just take the the side to be 1 and hence the surface area=6a^2=6. What is wrong with my thinking

Bunuel · Answer

When a cube with a side of 4 units is cut into cubes with a side of 1 unit you get 64 little cubes. Each of them will have the surface area of 6 square units, thus the surface area of all 64 little cubes will be 6*64 square units.

pandeyashwin · Answer

Surface area of larger cube = 6 * 4*4 = 96 
No of cubes of 1cm each = 4*4*4 = 64
surface area of 64 small cubes = 64 * 6 * 1*1 = 384

% Inc = (384-96) 100 / 96
= 300%

shashankism · Answer

Volume of a 4 cm cube = 4^3 = 64
Volume of a 1 cm cube = 1^3 = 1

Total number of 1 cm cubes which can be taken out of 4 cm cube after cutting = 64/1 = 64

Surface area of 4 cm cube = 6*4^2= 6*16

Surface are of 1 cm cube = 6*1^2 = 6 
Surface area of 64 1 cm cube = 6*64

Total increase in surface area = 6*64 - 6*16 = 6*48
%ge increase in surface area = 6*48/6*16 x 100 = 300%

Answer D

Vordhosbn · Answer

The volume of the cube is 64 and its surface area is 6*16 = 96. We can carve out 64 cubes of 1x1 out of the bigger/original cube. The surface are of the little cubes are 6(1x1) = 6. We got 64 of them --> 64*6. (64*6- 96)/96 * 10* = 300.

AndyRugger · Answer

I solved this question somewhat differently using graphic techniques.

Easiest to draw a cube and assign an X,Y,Z origin axes.

A cube has six faces. I assigned the Area "A" to each face, which is 64 cm^2. But as it turns out, you don't need to know the actual area. It cancels out.

Now imagine the cube sliced into smaller cubes of 1 cm^2 each. You can think of the cube being sliced three times along each planar direction, like a block of cheese being diced.

If you imagine the cube being pulled apart like a Slinky, you realize that there are now 3 new interior surface area planes in each of the cardinal directions, and each interior plane counts 2x because it's counting the surface area of two adjacent small cubes.

So the math is:

XZ direction: 2 Original + 3*2 New Planes = 8 Total

XY direction: 2 Original + 3*2 New Planes = 8 Total

YZ direction: 2 Original + 3*2 New Planes = 8 Total

So where the original cube has six total area "A" surface planes, the new disassembled cube has the equivalent of 8*3 = 24 "A" surface planes.

A little mental math shows that 24 = 4*6, or 300% increase.

(I actually got the question wrong at first because I didn't realize I needed to count each interior plane twice to account for the adjacent cube faces.)

If a cube with the length of the side of 4 cm is cut into smaller cube

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If a cube with the length of the side of 4 cm is cut into smaller cube

Prep Toolkit

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