Find the volume (in cm3) of the figure. Round to the nearest hundredth of a unit.
A rectangular solid has a small piece missing in its bottom half.
The edge between the top face and the back face is 4 cm.
The edge between the right face and the back face is 4 cm.
The edge between the right face and the bottom face is 8 cm.
A half-circle is cut out of the front face with its flat part, which measures 2 cm, along the bottom edge of the face. The space cut out from this half-circle continues straight through the block, such that another half-circle is cut out from the back face, and the space between is empty.

To find the volume of the given rectangular solid with a missing half-cylinder, let's break down the problem into steps:

### Step 1: Find the volume of the full rectangular solid.
The dimensions of the rectangular solid can be inferred from the description:
- The edge between the top face and the back face is the height, \( h = 4 \, \text{cm} \).
- The edge between the right face and the back face is the width, \( w = 4 \, \text{cm} \).
- The edge between the right face and the bottom face is the length, \( l = 8 \, \text{cm} \).

The volume of the full rectangular solid without the missing piece is calculated using the formula for the volume of a rectangular prism:
\[
V_{\text{rectangular}} = l \times w \times h
\]
\[
V_{\text{rectangular}} = 8 \, \text{cm} \times 4 \, \text{cm} \times 4 \, \text{cm} = 128 \, \text{cm}^3
\]

### Step 2: Find the volume of the missing half-cylinder.
- The diameter of the half-circle cut out from the front and back face is \( 2 \, \text{cm} \), so the radius of the half-circle is \( r = \frac{2}{2} = 1 \, \text{cm} \).
- The length of the half-cylinder is the same as the length of the block, which is \( l = 8 \, \text{cm} \).

The volume of a full cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
For the half-cylinder, the volume is half that of a full cylinder:
\[
V_{\text{half-cylinder}} = \frac{1}{2} \times \pi r^2 l
\]
Substitute the known values:
\[
V_{\text{half-cylinder}} = \frac{1}{2} \times \pi \times (1 \, \text{cm})^2 \times 8 \, \text{cm}
\]
\[
V_{\text{half-cylinder}} = \frac{1}{2} \times \pi \times 8 = 4\pi \, \text{cm}^3
\]
\[
V_{\text{half-cylinder}} \approx 4 \times 3.1416 = 12.57 \, \text{cm}^3
\]

### Step 3: Subtract the missing volume from the volume of the full rectangular solid.
Now, subtract the volume of the half-cylinder from the volume of the rectangular solid:
\[
V_{\text{total}} = V_{\text{rectangular}} - V_{\text{half-cylinder}}
\]
\[
V_{\text{total}} = 128 \, \text{cm}^3 - 12.57 \, \text{cm}^3 = 115.43 \, \text{cm}^3
\]

### Final Answer:
The volume of the figure is approximately \( 115.43 \, \text{cm}^3 \).

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Would you like any further clarification on this? Here are five related questions to deepen understanding:

1. How would the volume change if the diameter of the cut-out half-cylinder were increased?
2. How do you find the surface area of this figure?
3. What is the volume if the cut-out were a full cylinder instead of a half-cylinder?
4. How do you calculate the volume of a similar figure with an elliptical cut-out?
5. How does the orientation of the cut-out affect the volume calculation?

**Tip**: When subtracting volumes, make sure that the removed section's dimensions and position align with the object's total dimensions!

Find the volume (in cm3) of the figure. Round to the nearest hundredth of a unit. A rectangular solid has a small piece missing in its bottom half. The edge between the top face and the back face is 4 cm. The edge between the right face and the back face is 4 cm. The edge between the right face and the bottom face is 8 cm. A half-circle is cut out of the front face with its flat part, which measures 2 cm, along the bottom edge of the face. The space cut out from this half-circle continues straight through the block, such that another half-circle is cut out from the back face, and the space between is empty.

Discover how to calculate the volume of a rectangular solid with a half-cylinder cut-out. Using key geometric concepts and formulas, this problem involves determining the volume of both the solid and the missing half-cylinder. The solution is rounded to the nearest hundredth. Suitable for students in Grades 9-11.

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