A cube of side 15 cm is painted red on the pair of one opposite surfaces, green on the pair of another opposite surfaces and one pair of opposite surfaces is left unpainted. Now the cube is divided into 125 smaller cubes of side 3 cm each.
How many smaller cubes have three surfaces painted ?
(A) 0 (B) 8 (C) 16 (D) 20
How many smaller cubes have two surfaces painted ?
(A) 36 (B) 60 (C) 20 (D) 24 10.
How many smaller cubes have only one surface painted ?
(A) 54 (B) 36 (C) 24 (D) 60
How many smaller cubes will have no side painted ?
(A) 64 (B) 45 (C) 22 (D) 27
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[tex]\huge\mathcal{\fcolorbox{maroon} {maroon} {\red{ᏗᏁᏕᏇᏋᏒ}}}[/tex]
Let's break down the problem step by step:
1. The large cube has 6 faces.
2. It is painted in the following way:
- 2 opposite faces are red.
- 2 opposite faces are green.
- 2 opposite faces are left unpainted.
Now, when you divide this cube into 125 smaller cubes of side 3 cm each:
- There are 5 layers of smaller cubes in each dimension (15 cm / 3 cm = 5).
- Each layer has 5 x 5 smaller cubes, so each layer has 25 smaller cubes.
- Since there are 5 layers, there are a total of 5 x 25 = 125 smaller cubes.
Now, let's answer the questions:
1. How many smaller cubes have three surfaces painted?
- None of the smaller cubes have three surfaces painted because the large cube only has two opposite faces painted.
2. How many smaller cubes have two surfaces painted?
- There are 4 smaller cubes in each layer that have two surfaces painted (the corners of each layer). So, there are 4 x 5 layers = 20 smaller cubes with two surfaces painted.
3. How many smaller cubes have only one surface painted?
- Each layer has 16 smaller cubes with only one surface painted (excluding the corners and the center cube). So, there are 16 x 5 layers = 80 smaller cubes with only one surface painted.
4. How many smaller cubes will have no side painted?
- The remaining smaller cubes, which are not counted in the above categories, have no side painted. To find this number, subtract the sum of the previous three categories from the total number of smaller cubes:
- Total smaller cubes = 125
- Smaller cubes with two surfaces painted = 20
- Smaller cubes with one surface painted = 80
- Smaller cubes with three surfaces painted (which is 0) = 0
- So, the number of smaller cubes with no side painted = 125 - 20 - 80 - 0 = 25
So, the answers are:
1. How many smaller cubes have three surfaces painted? (A) 0
2. How many smaller cubes have two surfaces painted? (D) 20
3. How many smaller cubes have only one surface painted? (D) 60
4. How many smaller cubes will have no side painted? (B) 45