A solid is in the form of a right circular cone mounted on a hemisphere. The diameter of the base of the cone, which exactly coincides with hemisphere, is 7 cm and its height is 8 cm. The solid is placed in a cylindrical vessel of internal radius 7 cm and height 10 cm. How much water, in `cm^(3), `will be required to fill the vessel completely ?

Answer

To find out how much water is required to fill the cylindrical vessel completely, we need to calculate the volume of the cylinder and subtract the volume of the solid (which consists of a cone mounted on a hemisphere).

### Step-by-Step Solution:

1. **Calculate the Volume of the Cylinder:**
The formula for the volume of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Radius \( r = 7 \) cm
- Height \( h = 10 \) cm

Substituting the values:
\[ V = \pi (7)^2 (10) = \pi (49) (10) = 490\pi \text{ cm}^3 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = 490 \times \frac{22}{7} = 1540 \text{ cm}^3 \]

2. **Calculate the Volume of the Solid (Cone + Hemisphere):**
The volume of the solid consists of the volume of the cone and the volume of the hemisphere.

- **Volume of the Cone:**
The formula for the volume of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
Given:
- Diameter of the base of the cone = 7 cm, so radius \( r = \frac{7}{2} = 3.5 \) cm
- Height of the cone \( h = 8 \) cm

Substituting the values:
\[ V = \frac{1}{3} \pi (3.5)^2 (8) = \frac{1}{3} \pi (12.25)(8) = \frac{1}{3} \pi (98) = \frac{98\pi}{3} \text{ cm}^3 \]

- **Volume of the Hemisphere:**
The formula for the volume of a hemisphere is:
\[ V = \frac{2}{3} \pi r^3 \]
Using the same radius \( r = 3.5 \) cm:
\[ V = \frac{2}{3} \pi (3.5)^3 = \frac{2}{3} \pi (42.875) = \frac{85.75\pi}{3} \text{ cm}^3 \]

- **Total Volume of the Solid:**
Now, we add the volumes of the cone and the hemisphere:
\[ V_{\text{solid}} = \frac{98\pi}{3} + \frac{85.75\pi}{3} = \frac{(98 + 85.75)\pi}{3} = \frac{183.75\pi}{3} \text{ cm}^3 \]
Simplifying:
\[ V_{\text{solid}} = 61.25\pi \text{ cm}^3 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V_{\text{solid}} \approx 61.25 \times \frac{22}{7} \approx 192.5 \text{ cm}^3 \]

3. **Calculate the Amount of Water Required:**
The amount of water required to fill the vessel is the volume of the cylinder minus the volume of the solid:
\[ \text{Water Required} = V_{\text{cylinder}} - V_{\text{solid}} = 1540 - 192.5 = 1347.5 \text{ cm}^3 \]

### Final Answer:
The amount of water required to fill the vessel completely is **1347.5 cm³**.