Chapter 12.1, Problem 11PSC

a.

To determine

To calculate: The number of small cubes having six faces painted.

Answer to Problem 11PSC

There are no small cubes having six faces painted.

Explanation of Solution

Given information:

A six inch cube is painted on the outside and cut into 27 smaller cubes.

Calculation:

  

The number of small cubes having various faces painted are as follows:

6 faces painted = 0,

5 faces painted = 0,

4 faces painted = 0,

3 faces painted = 8

2 faces painted = 10,

1 faces painted = 6,

No faces painted = 1.

8 cubes like ⏸.

12 cubes like ⏸.

6 cubes like ⏸.

1 in center.

b.

To determine

To find: The probability that it has at least two painted faces if one of the small cubes is selected at random.

Answer to Problem 11PSC

The probability is $\frac{20}{27}$ .

Explanation of Solution

Given information:

A six inch cube is painted on the outside and cut into 27 smaller cubes.

Formula used:

Probability of event to happen is

  $P\left(E\right)=\frac{Numberoffavorableoutcomes}{TotalNumberofoutcomes}$

Calculation:

  

The number of small cubes having various faces painted are as follows:

6 faces painted = 0,

5 faces painted = 0,

4 faces painted = 0,

3 faces painted = 8

2 faces painted = 10,

1 faces painted = 6,

No faces painted= 1.

8 cubes like ⏸.

12 cubes like ⏸.

6 cubes like ⏸.

1 in center.

Probability means possibility.

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.

Probability of event to happen is

  $P\left(E\right)=\frac{Numberoffavorableoutcomes}{TotalNumberofoutcomes}$

A six inch cube is painted on the outside and cut into 27 smaller cubes.

Thera are 27 total number of outcomes.

The number of favorable outcomes $=8+12=20$

P(two painted faces if one of the small cubes is selected at random) $=\frac{20}{27}=74.07\%$ .

c.

To determine

To calculate: The total area of the unpainted surfaces.

Answer to Problem 11PSC

The total area of the unpainted surfaces is $432$ .

Explanation of Solution

Given information:

A six inch cube is painted on the outside and cut into 27 smaller cubes.

Formula used:

Total area = Area of each cube $\times$ Total number of unpainted faces

Calculation:

  

Each cube is 2 by 2, so the area of each cube = 4.

Type of cube	Number of unpainted faces	Number of cubes	Number of unpainted faces
⏸	3	8	24
⏸	4	12	48
⏸	5	6	30
In center	6	1	6

Total number of unpainted faces $=24+48+30+6=108$

Total area = Area of each cube $\times$ Total number of unpainted faces

Total area $=4\times 108=432$