To find the volume of the solid, we need to compute the volumes of the cylinder, cone, and hemisphere, and then sum them up.
### Step 1: Analyze the given data
- **Radius of the cylinder, cone, and hemisphere**: \( r = 3.5 \, \text{cm} \)
- **Height of the cylinder**: \( h_{\text{cyl}} = 6.5 \, \text{cm} \)
- **Total height of the solid**: \( 12.8 \, \text{cm} \)
Using the total height, the height of the cone is calculated as:
\[
h_{\text{cone}} = \text{Total height} - \text{Height of the cylinder} - \text{Radius of the hemisphere}
\]
\[
h_{\text{cone}} = 12.8 - 6.5 - 3.5 = 2.8 \, \text{cm}.
\]
### Step 2: Volume of the cylinder
The volume of a cylinder is given by:
\[
V_{\text{cyl}} = \pi r^2 h_{\text{cyl}}.
\]
Substituting the values:
\[
V_{\text{cyl}} = \pi (3.5)^2 (6.5) = \pi (12.25)(6.5) = \pi (79.625) \approx 250.2 \, \text{cm}^3.
\]
### Step 3: Volume of the cone
The volume of a cone is given by:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h_{\text{cone}}.
\]
Substituting the values:
\[
V_{\text{cone}} = \frac{1}{3} \pi (3.5)^2 (2.8) = \frac{1}{3} \pi (12.25)(2.8) = \frac{1}{3} \pi (34.3) \approx 35.9 \, \text{cm}^3.
\]
### Step 4: Volume of the hemisphere
The volume of a hemisphere is given by:
\[
V_{\text{hemi}} = \frac{2}{3} \pi r^3.
\]
Substituting the values:
\[
V_{\text{hemi}} = \frac{2}{3} \pi (3.5)^3 = \frac{2}{3} \pi (42.875) = \frac{2}{3} (134.6) \approx 89.8 \, \text{cm}^3.
\]
### Step 5: Total volume of the solid
The total volume of the solid is the sum of the volumes of the cylinder, cone, and hemisphere:
\[
V_{\text{total}} = V_{\text{cyl}} + V_{\text{cone}} + V_{\text{hemi}}.
\]
Substituting the values:
\[
V_{\text{total}} = 250.2 + 35.9 + 89.8 \approx 375.9 \, \text{cm}^3.
\]
Rounding to the nearest integer:
\[
V_{\text{total}} \approx 376 \, \text{cm}^3.
\]
### Final Answer:
The volume of the solid is approximately **376 cm³**. The given options do not match the computed value exactly, but the closest match is **375 cm³** (Option B).
