The internal and external diameters of a hollow hemispherical vessel are 21 cm and 28 cm respectively. Find :
volume of material of the vessel.

Answer

To find the volume of the material of the hollow hemispherical vessel, we will follow these steps:

### Step 1: Determine the radii
The internal diameter of the vessel is 21 cm, and the external diameter is 28 cm. We can find the radii by dividing the diameters by 2.

- Internal radius (r) = 21 cm / 2 = 10.5 cm
- External radius (R) = 28 cm / 2 = 14 cm

### Step 2: Use the formula for the volume of a hemisphere
The formula for the volume \( V \) of a hemisphere is given by:

\[ V = \frac{2}{3} \pi r^3 \]

### Step 3: Calculate the volume of the outer hemisphere
Using the external radius \( R = 14 \) cm, we calculate the volume of the outer hemisphere:

\[ V_{outer} = \frac{2}{3} \pi R^3 = \frac{2}{3} \pi (14)^3 \]

Calculating \( 14^3 \):

\[ 14^3 = 2744 \]

Thus,

\[ V_{outer} = \frac{2}{3} \pi (2744) = \frac{5488}{3} \pi \]

### Step 4: Calculate the volume of the inner hemisphere
Using the internal radius \( r = 10.5 \) cm, we calculate the volume of the inner hemisphere:

\[ V_{inner} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (10.5)^3 \]

Calculating \( 10.5^3 \):

\[ 10.5^3 = 1157.625 \]

Thus,

\[ V_{inner} = \frac{2}{3} \pi (1157.625) = \frac{2315.25}{3} \pi \]

### Step 5: Find the volume of the material of the vessel
The volume of the material of the vessel is the difference between the volume of the outer hemisphere and the volume of the inner hemisphere:

\[ V_{material} = V_{outer} - V_{inner} \]

Substituting the values we calculated:

\[ V_{material} = \left(\frac{5488}{3} \pi - \frac{2315.25}{3} \pi\right) \]

Combining the fractions:

\[ V_{material} = \frac{5488 - 2315.25}{3} \pi = \frac{3172.75}{3} \pi \]

### Step 6: Calculate the numerical value
Using \( \pi \approx 3.14 \):

\[ V_{material} \approx \frac{3172.75}{3} \times 3.14 \approx 3323.83 \text{ cm}^3 \]

Thus, the volume of the material of the vessel is approximately **3323.83 cm³**.

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