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# 1. A cube is painted red on all six sides, then cut into 125 smaller congruent cubes. How many of the small cubes are painted red on ex

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1. A cube is painted red on all six sides, then cut into 125 smaller congruent cubes. How many of the small cubes are painted red on exactly one side?

2. A cube is painted red on all six sides, then cut into 125 smaller congruent cubes. How many of the small cubes are not painted red on any sides?

3. A cube is painted red on all six sides, then cut into 125 smaller congruent cubes. A small cube is selected at random and rolled. What is the probability that it has a red side facing up?

Guest Apr 2, 2015

#1
+17746
+10

To cut it into 125 congruent smaller cubes requires 5 horizontal cuts and 5 vertical cuts.

The only cubes that are painted red on one side are the interior cubes on each side. Of the 25 cubes on each side, nine of them are painted red on only one side. 9 x 6 = 54 cubes with red on only one side (9 on each side x 6 sides).

The only cubes that are not painted at all are the interior cubes; there are 3 x 3 x 3 = 27 interior cubes.

Of all the 125 cubes,

27 are interior cubes, that have no sides painted red.

54 are cubes that are painted red on only one side.

36 are cubes that are painted red on two sides.

8 are corner cubes; they are painted red on three sides.

The probability of a red side facing up:  (27/125)(0/6) + (54/125)(1/6) + (36/125)(2/6) + (8/125)(3/6)

(The first set of parentheses contains the probability of that type of cube is chosen; the second set of parentheses contains the probability of having a red side facing up; add together all these individual probabilities.)

... finish the multiplying and adding ...

geno3141  Apr 2, 2015
#1
+17746
+10

To cut it into 125 congruent smaller cubes requires 5 horizontal cuts and 5 vertical cuts.

The only cubes that are painted red on one side are the interior cubes on each side. Of the 25 cubes on each side, nine of them are painted red on only one side. 9 x 6 = 54 cubes with red on only one side (9 on each side x 6 sides).

The only cubes that are not painted at all are the interior cubes; there are 3 x 3 x 3 = 27 interior cubes.

Of all the 125 cubes,

27 are interior cubes, that have no sides painted red.

54 are cubes that are painted red on only one side.

36 are cubes that are painted red on two sides.

8 are corner cubes; they are painted red on three sides.

The probability of a red side facing up:  (27/125)(0/6) + (54/125)(1/6) + (36/125)(2/6) + (8/125)(3/6)

(The first set of parentheses contains the probability of that type of cube is chosen; the second set of parentheses contains the probability of having a red side facing up; add together all these individual probabilities.)

... finish the multiplying and adding ...

geno3141  Apr 2, 2015
#2
+93025
0

Very nice, geno......you visualize in 3-D WAY better than I do....!!!

I'll recommend this one for the Wrap....

CPhill  Apr 2, 2015
#3
+94202
+5

It turns out that this is a double post:

other one

http://web2.0calc.com/questions/a-5x5x5-cube-is-formed-by-assembling-125-unit-cubes-nine-unit-squares-are-painted-on-each-of-the-six-faces-of-the-cube-according-to-the-pat

Do not double post like this