A wooden cube whose edge length is 10 inches is composed of

Question

A wooden cube whose edge length is 10 inches is composed of smaller cubes with edge lengths of one inch. The outside surface of the large cube is painted red and then it is split up into its smaller cubes. If one cube is randomly selected from the small cubes, what is the probability that the cube will have AT LEAST one red face?

A. 36.0%
B. 48.8%
C. 50.0%
D. 52.5%
E. 60%

Can someone elaborate on the approach to tackle a problem such as this one?

A. 36.0%
B. 48.8%
C. 50.0%
D. 52.5%
E. 60%

Can someone elaborate on the approach to tackle a problem such as this one?

Bunuel · Accepted Answer

The cube is composed of 10^3 little cubes. Out of them 8^3 non-exposed central cubes won't be colored at all.

Thus the probability is P(at least one red face) = 1 - P(no red face) = 1 - 8^3/10^3= 488/10^3.

Answer: B.

PatrickS · Answer

This was my approach but it took me at least 5 min to figure it out, anyone can think of a quicker way?

Step 1: Calculate the total number of cubes
Volume = L*W*H=10*10*10=1000 cubes in total

Step 2: Calculate the number of cubes that are NOT painted:
Volume=L*W*H=8*8*8= 512 - took 8 since on each 2D slice we will have the first and last cube painted

Step 3: Calculate the number of cubes that will have at least 1 side painted:
#Cubes with at least 1 side painted = 1000-512=488

Step 4: Calculate probability
P(Cubes with at least 1 side) =  \frac{#Cubes with at least 1 side painted}{Total number of cubes} =  \frac{488}{1000} =48.8%

Bunuel · Answer

Similar questions to practice: the-entire-exterior-of-a-large-wooden-cube-is-painted-red-155955.html if-a-4-cm-cube-is-cut-into-1-cm-cubes-then-what-is-the-107843.html a-big-cube-is-formed-by-rearranging-the-160-coloured-and-99424.html a-large-cube-consists-of-125-identical-small-cubes-how-110256.html 64-small-identical-cubes-are-used-to-form-a-large-cube-151009.html Hope it helps.

kundankshrivastava · Answer

Total number of small cubes = 10^3 = 1000

All the cubes which are in side the outer layer of cubes have no side colored.

Number of cubes inside the outer layer = 8*8*8 = 512

probability that the cube will have AT LEAST one red face = 1- probability that the cube will have no red face

=1-(512/1000) = 1-0.512 = 0.488 = 48.8%

Answer is B

ronr34 · Answer

Hi Bunuel,
I tried this with the counting method:
8*8 squares in the inside of every side of the cube.
now we choose one side, and what's left there is 39 cubes. same for that exact opposite side.
now all that's left is to connect the two sides (8 cubes to connect * 4 edges to connect)
I get to 494/1000. where am I going wrong?

Bunuel · Answer

How is there 39 cubes left? Does 39 and 8^2 total 100?

samirchaudhary · Answer

Though this question can be done better the way already discussed, Let's count the number of cubes with 1-face, 2-faces and 3-faces painted. This might give us a better picture if the question is modifies as : probability of getting a cube with exactly 2 faces painted or exactly 3 faces painted.

1-face painted:

Cubes with one face painted lies on each face except along the edges of the face.
So total number of 1-face painted cubes = 8*8*6 = 384<when we exclude edges, we are left with 8*8><6 for 6 faces>

2-faces painted:
Such cubes exist along the edges except the ones at the corners. And total number of edges = 12
So, total number of 2-faces painted cubes = 8*12 = 96 <Out of 10 cubes along an edge, only 8 are 2-faces painted, the ones at the corners are all three faces painted>

3-faces painted:
The ones at the corners are 3-faces painted. Number of corners = 8
Total number of 3-faces painted cubes= 8.

AnisMURR · Answer

We have in total 10*10*10 = 1000 cubes.

Let s count the zero no red face and the substract it from 1000.

The number of cubes with no red face are 8*8*8=512 it is a cube inscribed in the biggest cube with a side equal to 8 inches.

So 1000-512= 488... it gives 48,8%

Answer B

PreiteeRanjan · Answer

their are in total 1000 cubes possible of smaller kind,10*10*10
number of untouched cube volume =(N-2)^3
So probability =1-(8^3/10^3)=48.8%

saby1410 · Answer

VeritasKarishma  
Please explain this question i didn't get why 2 cubes will be deduced and 8 will left 
Pls explain this question

Posted from my mobile device

saby1410 · Answer

VeritasKarishma  
Please explain this question i didn't get why 2 cubes will be deduced and 8 will left 
Pls explain this question

Posted from my mobile device

KarishmaB · Answer

saby1410 
The big cube of 10 inches side is made up of little cubes of 1 inch side. So you place 10 little cubes in a row side by side, then another 10 cube row next to it, then another till you get a square of 100 cubes. Next you place another 100 cube square on top of it, then another till you get a big cube of 1000 little cubes (10 by 10 by 10). Think of a 10 by 10 rubik's cube.

Now you paint the outer surface. So every cube that makes up the outer surface will have at least one side painted. The cubes lying inside after removing the cubes on the outside (one on each side) will be 8*8*8. These will make up 512 cubes so 488 cubes are painted.

Probability = 488/1000 * 100 = 48.8%

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A wooden cube whose edge length is 10 inches is composed of

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