To find the number of cubes with side lengths of \( \frac{1}{4} \) cm that can completely fill the rectangular prism, we first need to calculate the volume of the prism and then the volume of one of the cubes.
Step 1: Calculate the volume of the rectangular prism.
The formula for the volume \( V \) of a rectangular prism is given by:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given dimensions of the prism:
- Length: \( 3 \frac{3}{4} \) cm = \( 3.75 \) cm
- Width: \( 1 \) cm
- Height: \( 2 \frac{1}{4} \) cm = \( 2.25 \) cm
Now, we can calculate the volume:
\[ V = 3.75 , \text{cm} \times 1 , \text{cm} \times 2.25 , \text{cm} \]
Calculating step-by-step:
- Calculate \( 3.75 \times 1 = 3.75 \)
- Then calculate \( 3.75 \times 2.25 \):
\[ 3.75 \times 2.25 = \frac{375}{100} \times \frac{225}{100} = \frac{375 \times 225}{10000} \]
We calculate \( 375 \times 225 \):
\[ 375 \times 225 = 84375 \] So then:
\[ V = \frac{84375}{10000} = 8.4375 \text{ cm}^3 \]
Step 2: Calculate the volume of one cube.
The volume \( v \) of a cube is given by:
\[ v = \text{side}^3 \]
Given the side length of the cube:
\[ \text{side} = \frac{1}{4} \text{ cm} \]
Calculating the volume:
\[ v = \left(\frac{1}{4}\right)^3 = \frac{1}{64} \text{ cm}^3 \]
Step 3: Calculate the number of cubes that fill the prism.
To find the number of cubes, divide the volume of the prism by the volume of one cube:
\[ \text{Number of cubes} = \frac{V}{v} = \frac{8.4375 \text{ cm}^3}{\frac{1}{64} \text{ cm}^3} \]
This can be rewritten as:
\[ \text{Number of cubes} = 8.4375 \times 64 \]
Calculating \( 8.4375 \times 64 \):
\[ 8.4375 \times 64 = 540 \]
Thus, the number of cubes with side lengths of \( \frac{1}{4} \) cm that can completely fill the prism is
\[ \boxed{540} \]