Question

The three-dimensional figure below is a solid rectangular prism with a hole in the shape of another rectangular prism going through the center of it. Find the volume of the solid in cubic millimeters.

A solid rectangular prism with a hole in the shape of another rectangular prism going through the center of it is shown. The rectangular prism is 15 millimeters long, 6 millimeters wide, and 6 millimeters high. The rectangular prism-shaped-hole is 15 millimeters long, 4 millimeters wide, and 4 millimeters high.

Answer 1

To find the volume of the solid rectangular prism with a hole, we will first calculate the volume of the outer solid prism and then subtract the volume of the inner hole.

1. **Volume of the outer solid prism:**

The formula for the volume of a rectangular prism is given by:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]

For the outer solid prism:
- Length = 15 mm
- Width = 6 mm
- Height = 6 mm

Calculating the volume:
\[
\text{Volume}_{\text{outer}} = 15 \, \text{mm} \times 6 \, \text{mm} \times 6 \, \text{mm} = 540 \, \text{mm}^3
\]

2. **Volume of the hole (inner prism):**

Using the same formula for the volume of the rectangular prism for the hole:
- Length = 15 mm
- Width = 4 mm
- Height = 4 mm

Calculating the volume of the hole:
\[
\text{Volume}_{\text{hole}} = 15 \, \text{mm} \times 4 \, \text{mm} \times 4 \, \text{mm} = 240 \, \text{mm}^3
\]

3. **Volume of the solid with the hole:**

Now, subtract the volume of the hole from the volume of the outer prism:
\[
\text{Volume}_{\text{solid}} = \text{Volume}_{\text{outer}} - \text{Volume}_{\text{hole}}
\]
\[
\text{Volume}_{\text{solid}} = 540 \, \text{mm}^3 - 240 \, \text{mm}^3 = 300 \, \text{mm}^3
\]

Thus, the volume of the solid rectangular prism with the hole is:
\[
\boxed{300} \, \text{mm}^3
\]