To find the volume of this figure, we need to calculate the volume of the rectangular prism first, and then subtract the volume of the semicircular cylinder that has been subtracted from it.

Step 1: Volume of the Rectangular Prism

The formula for the volume of a rectangular prism is: $V = \text{length} \times \text{width} \times \text{height}$ From the diagram:

• Length = 8 cm
• Width = 4 cm
• Height = 4 cm

So, $V_{\text{prism}} = 8 \times 4 \times 4 = 128 \, \text{cm}^3$

Step 2: Volume of the Semicircular Cylinder

The volume of a full cylinder is calculated using: $V = \pi \times \text{radius}^2 \times \text{height}$ However, since we have a semicircular cylinder (half of a cylinder), we divide the result by 2.

• Radius = 2 cm (from the 4 cm diameter shown in the figure)
• Height of the semicircular cylinder = 8 cm (same as the length of the prism)

Thus, $V_{\text{cylinder}} = \frac{1}{2} \times \pi \times (2)^2 \times 8$ $= \frac{1}{2} \times \pi \times 4 \times 8$ $= \frac{1}{2} \times \pi \times 32$ $= 16\pi$

Using $\pi \approx 3.14159$, $V_{\text{cylinder}} \approx 16 \times 3.14159 = 50.27 \, \text{cm}^3$

Step 3: Subtract the Volume of the Semicircular Cylinder from the Prism

$V_{\text{figure}} = V_{\text{prism}} - V_{\text{cylinder}}$ $V_{\text{figure}} = 128 - 50.27 = 77.73 \, \text{cm}^3$

Answer

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

Would you like further clarification on this solution?

Here are 5 related questions:

1. How would the volume change if the diameter of the semicircular cutout was increased?
2. What is the formula for the volume of a quarter-cylinder, and how might it apply to different shapes?
3. How does changing the height of the rectangular prism affect the volume calculation?
4. How can the surface area of this shape be calculated?
5. What is the impact of rounding $\pi$ to different decimal places on volume calculations?

Tip: Always check if the cut-out shape (like the semicircular cylinder) removes volume when calculating the final answer for combined shapes.