Question

Consider the rectangular prism with base of 3 3/4 cm width is 1 cm and height is 2 1/4 cm. How many cubes with side lengths of 1/4 cm completely fill the prism?

Answer 1

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:

1. Calculate \( 3.75 \times 1 = 3.75 \)
2. 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

\[
\boxed{540}
\]