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!