A solid is in the form of a cone standing on a hemisphere with both their radii being equal to 8 cm and the height of cone is equal to its radius . Find the volume of solid.

Answer

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

### Step 1: Identify the dimensions
Given:
- Radius of the cone (r) = 8 cm
- Height of the cone (h) = 8 cm (since the height is equal to the radius)

### Step 2: Calculate the volume of the cone
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V_{\text{cone}} = \frac{1}{3} \pi (8)^2 (8) \]
Calculating:
\[ V_{\text{cone}} = \frac{1}{3} \pi (64) (8) = \frac{1}{3} \pi (512) = \frac{512}{3} \pi \text{ cm}^3 \]

### 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 values:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (8)^3 \]
Calculating:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (512) = \frac{1024}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the total volume of the solid
Now, we add the volumes of the cone and the hemisphere:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} \]
Substituting the values:
\[ V_{\text{total}} = \frac{512}{3} \pi + \frac{1024}{3} \pi \]
Combining the fractions:
\[ V_{\text{total}} = \frac{512 + 1024}{3} \pi = \frac{1536}{3} \pi = 512 \pi \text{ cm}^3 \]

### Step 5: Calculate the numerical value
Using \(\pi \approx \frac{22}{7}\):
\[ V_{\text{total}} \approx 512 \times \frac{22}{7} \approx 512 \times 3.14 \approx 1607.68 \text{ cm}^3 \]

### Final Answer
The volume of the solid is approximately \(1607.68 \text{ cm}^3\).

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