The image shows a problem asking for the volume of a combined solid shape. The solid is a combination of a rectangular prism and a half-cylinder on top of it. Let's break down the calculation to find the total volume.

Dimensions provided:

• Rectangular prism:

  • Length: 12 cm
  • Width: 7 cm
  • Height: 2 cm

• Half-cylinder:

  • Length (same as the prism's length): 12 cm
  • Radius: 7/2 = 3.5 cm (since the width of the prism is the diameter of the cylinder)
  • Height (same as the prism's width): 7 cm

Step 1: Volume of the Rectangular Prism

The volume $V_{\text{prism}}$ of a rectangular prism is calculated using the formula: $V_{\text{prism}} = \text{length} \times \text{width} \times \text{height}$ $V_{\text{prism}} = 12 \times 7 \times 2 = 168 \, \text{cm}^3$

Step 2: Volume of the Half-Cylinder

The volume $V_{\text{cylinder}}$ of a full cylinder is calculated using the formula: $V_{\text{cylinder}} = \pi \times \text{radius}^2 \times \text{height}$ Since we have a half-cylinder, the volume $V_{\text{half-cylinder}}$ is half of the full cylinder's volume: $V_{\text{half-cylinder}} = \frac{1}{2} \times \pi \times \left(\frac{7}{2}\right)^2 \times 12$ $V_{\text{half-cylinder}} = \frac{1}{2} \times \pi \times 3.5^2 \times 12$ $V_{\text{half-cylinder}} = \frac{1}{2} \times \pi \times 12.25 \times 12$ $V_{\text{half-cylinder}} = \frac{1}{2} \times \pi \times 147$ $V_{\text{half-cylinder}} \approx \frac{1}{2} \times 3.1416 \times 147 \approx 230.907 \, \text{cm}^3$

Step 3: Total Volume of the Combined Solid

The total volume $V_{\text{total}}$ of the combined solid is the sum of the volumes of the rectangular prism and the half-cylinder: $V_{\text{total}} = V_{\text{prism}} + V_{\text{half-cylinder}}$ $V_{\text{total}} = 168 + 230.907 \approx 398.91 \, \text{cm}^3$

Final Answer

The volume of the combined solid is approximately 398.91 cm³.

Would you like more details on any part of the calculation, or do you have any other questions?

Here are some related questions you might find interesting:

1. What would be the volume if the half-cylinder were replaced by a full cylinder?
2. How would the volume change if the dimensions of the rectangular prism were doubled?
3. Can you calculate the surface area of this combined solid?
4. What is the volume of the combined solid if the half-cylinder was instead a quarter-cylinder?
5. How does the volume of the combined solid compare to a solid composed of just the full cylinder?

Tip: Understanding the formulas for basic geometric shapes like cylinders and prisms is essential for solving combined solid problems quickly and accurately.