A cone of height 15 cm and diameter 7 cm is mounted on a hemisphere of same diameter. Determine the volume of the solid thus formed.

Answer

To find the volume of the solid formed by a cone mounted on a hemisphere, we can follow these steps:

### Step 1: Identify the dimensions of the cone and hemisphere
- Height of the cone (h) = 15 cm
- Diameter of the cone = 7 cm
- Radius of the cone (r) = Diameter / 2 = 7 cm / 2 = 3.5 cm

Since the hemisphere has the same diameter, its radius is also:
- Radius of the hemisphere = 3.5 cm

### Step 2: Calculate the volume of the cone
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V_{\text{cone}} = \frac{1}{3} \pi (3.5)^2 (15) \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now substituting back:
\[ V_{\text{cone}} = \frac{1}{3} \pi (12.25)(15) \]
Calculating \( 12.25 \times 15 \):
\[ 12.25 \times 15 = 183.75 \]
So,
\[ V_{\text{cone}} = \frac{1}{3} \pi (183.75) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times 183.75 \]
Calculating:
\[ V_{\text{cone}} = \frac{22 \times 183.75}{21} = \frac{4042.5}{21} \approx 192.5 \text{ cm}^3 \]

### Step 3: Calculate the volume of the hemisphere
The formula for the volume of a hemisphere is:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
Substituting the radius:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (3.5)^3 \]
Calculating \( (3.5)^3 \):
\[ (3.5)^3 = 42.875 \]
Now substituting back:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (42.875) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V_{\text{hemisphere}} = \frac{2}{3} \times \frac{22}{7} \times 42.875 \]
Calculating:
\[ V_{\text{hemisphere}} = \frac{2 \times 22 \times 42.875}{21} = \frac{1883.5}{21} \approx 89.7 \text{ cm}^3 \]

### Step 4: Calculate the total volume of the solid
Now, we can find the total volume of the solid:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} \]
Substituting the values:
\[ V_{\text{total}} = 192.5 + 89.7 \approx 282.2 \text{ cm}^3 \]

### Final Answer
The volume of the solid formed by the cone mounted on the hemisphere is approximately:
\[ \boxed{282.2 \text{ cm}^3} \]