The shape in the image is a composite solid made up of a rectangular prism and a half-cylinder. To find the total volume, we'll calculate the volume of each component and sum them up.

Given dimensions:

• The rectangular part has a width of $8$ units, a length of $16$ units, and a height equal to the radius of the half-cylinder, which is $4$ units (since the diameter of the half-cylinder is $8$, the radius is $4$).
• The half-cylinder has a radius $r = 4$ units and a length $L = 16$ units.

Step 1: Volume of the rectangular prism

The volume $V_{\text{rect}}$ of a rectangular prism is given by: $V_{\text{rect}} = \text{length} \times \text{width} \times \text{height}$ Substituting the values: $V_{\text{rect}} = 16 \times 8 \times 4 = 512 \text{ cubic units}.$

Step 2: Volume of the half-cylinder

The volume $V_{\text{cyl}}$ of a full cylinder is given by: $V_{\text{cyl}} = \pi r^2 h$ Since we have a half-cylinder, we divide the result by 2. The height $h$ of the cylinder is the same as the length $L$ of the rectangular prism, which is $16$ units.

Substituting the values: $V_{\text{half-cyl}} = \frac{1}{2} \times \pi \times 4^2 \times 16 = \frac{1}{2} \times \pi \times 16 \times 16 = \frac{1}{2} \times \pi \times 256 = 128\pi \text{ cubic units}.$ Approximating $\pi \approx 3.1416$: $V_{\text{half-cyl}} \approx 128 \times 3.1416 = 402.12 \text{ cubic units}.$

Step 3: Total volume

Adding the volumes of the rectangular prism and the half-cylinder: $V_{\text{total}} = V_{\text{rect}} + V_{\text{half-cyl}} = 512 + 402.12 = 914.12 \text{ cubic units}.$

Final Answer:

The total volume of the figure is approximately 914.12 cubic units.

Would you like more details or further clarification? Here are 5 related questions you might find useful:

1. How do you calculate the surface area of this composite figure?
2. What is the volume of a full cylinder with the same dimensions?
3. How would the volume change if the dimensions were doubled?
4. How do you convert this volume into liters (if working in units of centimeters)?
5. How do different composite shapes impact volume calculation?

Tip: Always break down composite shapes into simpler solids for easier volume or surface area calculations.