To solve for the volume of the figure, let's analyze it in two parts. The figure seems to be a rectangular prism with a semicircular cut-out on one side.

Step 1: Volume of the Rectangular Prism

The rectangular prism has dimensions:

• Length $L = 8 \, \text{cm}$
• Width $W = 4 \, \text{cm}$
• Height $H = 4 \, \text{cm}$

The volume of a rectangular prism is given by: $V_{\text{prism}} = L \times W \times H$ $V_{\text{prism}} = 8 \times 4 \times 4 = 128 \, \text{cm}^3$

Step 2: Volume of the Semicircular Cut-Out

The cut-out is a semicircular cylinder with:

• Radius $r = 2 \, \text{cm}$ (half of the width)
• Length (same as the prism's length) $L = 8 \, \text{cm}$

The volume of a full cylinder is given by: $V_{\text{cylinder}} = \pi r^2 L$ Since it's a semicircular cylinder, we take half of this volume: $V_{\text{cut-out}} = \frac{1}{2} \pi r^2 L$ $V_{\text{cut-out}} = \frac{1}{2} \pi (2)^2 \times 8$ $V_{\text{cut-out}} = \frac{1}{2} \pi \times 4 \times 8$ $V_{\text{cut-out}} = 16 \pi \approx 50.27 \, \text{cm}^3$

Step 3: Subtract the Cut-Out Volume from the Prism Volume

$V_{\text{total}} = V_{\text{prism}} - V_{\text{cut-out}}$ $V_{\text{total}} = 128 - 50.27$ $V_{\text{total}} \approx 77.73 \, \text{cm}^3$

Final Answer

The volume of the figure, rounded to the nearest hundredth, is approximately: $\boxed{77.73 \, \text{cm}^3}$

Would you like more details on any part of this solution?

Here are some related questions:

1. How would the answer change if the cut-out were a full circle instead of a semicircle?
2. What is the surface area of this figure after the cut-out is made?
3. How does the volume calculation differ if the dimensions are in meters instead of centimeters?
4. What would the volume be if the prism's length were doubled?
5. How would you calculate the volume if the cut-out shape were a triangle?

Tip: Always double-check unit consistency when calculating volume, especially if multiple shapes are involved.