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Answer :
To solve the question of how many smaller cubes can fit into a larger 1-inch cube, we'll proceed step by step for each case:
### 4. Cube with Edge [tex]\( \frac{1}{3} \)[/tex] inch
1. Determine the volume of the 1-inch cube:
- Volume = [tex]\( 1 \times 1 \times 1 = 1 \)[/tex] cubic inch.
2. Determine the volume of the smaller cube with edge [tex]\( \frac{1}{3} \)[/tex] inch:
- Volume = [tex]\( \left(\frac{1}{3}\right) \times \left(\frac{1}{3}\right) \times \left(\frac{1}{3}\right) = \left(\frac{1}{3}\right)^3 \)[/tex].
3. Calculate the volume:
- Volume = [tex]\( \frac{1}{27} \)[/tex] cubic inch.
4. Determine how many of these smaller cubes fit into the 1-inch cube:
- Count = [tex]\( \frac{1}{\frac{1}{27}} = 27 \)[/tex].
So, 27 cubes of edge [tex]\( \frac{1}{3} \)[/tex] inch can fill a 1-inch cube.
### 5. Cube with Edge [tex]\( \frac{1}{4} \)[/tex] inch
1. Volume of the smaller cube with edge [tex]\( \frac{1}{4} \)[/tex] inch:
- Volume = [tex]\( \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = \left(\frac{1}{4}\right)^3 \)[/tex].
2. Calculate the volume:
- Volume = [tex]\( \frac{1}{64} \)[/tex] cubic inch.
3. Determine how many of these smaller cubes fit into the 1-inch cube:
- Count = [tex]\( \frac{1}{\frac{1}{64}} = 64 \)[/tex].
So, 64 cubes of edge [tex]\( \frac{1}{4} \)[/tex] inch can fill a 1-inch cube.
These calculations show that 27 smaller cubes of edge [tex]\( \frac{1}{3} \)[/tex] inch and 64 smaller cubes of edge [tex]\( \frac{1}{4} \)[/tex] inch can completely and perfectly fit into a 1-inch cube.
### 4. Cube with Edge [tex]\( \frac{1}{3} \)[/tex] inch
1. Determine the volume of the 1-inch cube:
- Volume = [tex]\( 1 \times 1 \times 1 = 1 \)[/tex] cubic inch.
2. Determine the volume of the smaller cube with edge [tex]\( \frac{1}{3} \)[/tex] inch:
- Volume = [tex]\( \left(\frac{1}{3}\right) \times \left(\frac{1}{3}\right) \times \left(\frac{1}{3}\right) = \left(\frac{1}{3}\right)^3 \)[/tex].
3. Calculate the volume:
- Volume = [tex]\( \frac{1}{27} \)[/tex] cubic inch.
4. Determine how many of these smaller cubes fit into the 1-inch cube:
- Count = [tex]\( \frac{1}{\frac{1}{27}} = 27 \)[/tex].
So, 27 cubes of edge [tex]\( \frac{1}{3} \)[/tex] inch can fill a 1-inch cube.
### 5. Cube with Edge [tex]\( \frac{1}{4} \)[/tex] inch
1. Volume of the smaller cube with edge [tex]\( \frac{1}{4} \)[/tex] inch:
- Volume = [tex]\( \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = \left(\frac{1}{4}\right)^3 \)[/tex].
2. Calculate the volume:
- Volume = [tex]\( \frac{1}{64} \)[/tex] cubic inch.
3. Determine how many of these smaller cubes fit into the 1-inch cube:
- Count = [tex]\( \frac{1}{\frac{1}{64}} = 64 \)[/tex].
So, 64 cubes of edge [tex]\( \frac{1}{4} \)[/tex] inch can fill a 1-inch cube.
These calculations show that 27 smaller cubes of edge [tex]\( \frac{1}{3} \)[/tex] inch and 64 smaller cubes of edge [tex]\( \frac{1}{4} \)[/tex] inch can completely and perfectly fit into a 1-inch cube.
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