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Shtirlitz [24]
4 years ago
7

A 3x3x4 cuboid is painted blue and cut into 1x1 cubes. what is the expected value for the painted sides of a randomly selected c

ube?
Mathematics
1 answer:
Kobotan [32]4 years ago
7 0
A 3 x 3 x 4 cuboid can be cut into 3(3)(4) = 36 pieces of 1 x 1 cubes.

Out of the 36 pieces of 1 x 1 cubes, there are no 1 x 1 cube with the four sides painted, there are eight 1 x 1 cubes with only 3 sides painted, there are twenty 1 x 1 cubes with only 2 sides painted, there are 6 cubes with only 1 side painted and there are 2 cubes with no side painted.

The probability that the four sides of a randomly selected cube is painted is 0. The probability that only 3 sides of a randomly selected cube is painted is 8 / 36 = 2 / 9. The probability that only 2 sides of a randomly selected cube is painted is 20 / 36 = 5 / 9. The probability that only 1 side of a randomly selected cube is painted is 6 / 36 = 1 / 6. The probability that no side of a randomly selected cube is painted is 2 / 36 = 1 / 18.

Therefore, the expected value for the painted sides of a randomly selected cube is given by 4(0) + 3(2 / 9) + 2(5 / 9) + 1(1 / 6) + 0( 1 / 18) = 2 / 3 + 10 / 9 + 1 / 6 = 35 / 18 = 1.94
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U
=
(
−
2
,
1
)
u
=
(
-
2
,
1
)
,
v
=
(
5
,
−
4
)
v
=
(
5
,
-
4
)
The equation for finding the angle between two vectors
θ
θ
states that the dot product of the two vectors equals the product of the magnitudes of the vectors and the cosine of the angle between them.
u
⋅
v
=
|
u
|
|
v
|
c
o
s
(
θ
)
u
⋅
v
=
|
u
|
|
v
|
c
o
s
(
θ
)
Solve the equation for
θ
θ
.
θ
=
a
r
c
⋅
c
o
s
(
u
⋅
v
|
u
|
|
v
|
)
θ
=
a
r
c
⋅
c
o
s
(
u
⋅
v
|
u
|
|
v
|
)
Find the dot product of the vectors.
Tap for more steps...
−
14
-
14
Find the magnitude of
u
u
.
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√
5
5
Find the magnitude of
v
v
.
Tap for more steps...
√
41
41
Substitute the values into the equation for the angle between the vectors.
θ
=
arccos
⎛
⎜
⎜
⎝
−
14
(
√
5
)
⋅
(
√
41
)
⎞
⎟
⎟
⎠
θ
=
arccos
(
-
14
(
5
)
⋅
(
41
)
)
Simplify.
Tap for more steps...
2.93049932
2.93049932
u
=
(
−
2
,
1
)
,
v
=
(
5
,
−
4
)
u
=
(
-
2
,
1
)
,
v
=
(
5
,
-
4
)
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