A solid spherical ball of radius 21 cm is melted to form 'n' solid hemispherical bowl. If radius of each hemispherical bowl is half of radius of spherical ball, then find total surface area of all hemispherical bowls.

Options

A. 15846 `cm^(2)`
B. 16632 `cm^(2)`
C. 16028 `cm^(2)`
D. 16272 `cm^(2)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Find the volume of the solid spherical ball.
The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

Given the radius \( r \) of the spherical ball is 21 cm, we can substitute this value into the formula:

\[ V = \frac{4}{3} \pi (21)^3 \]

### Step 2: Calculate \( (21)^3 \).
Calculating \( 21^3 \):

\[ 21^3 = 21 \times 21 \times 21 = 441 \times 21 = 9261 \]

### Step 3: Substitute back into the volume formula.
Now substituting \( 21^3 \) back into the volume formula:

\[ V = \frac{4}{3} \pi (9261) = \frac{37044}{3} \pi \]

### Step 4: Find the radius of each hemispherical bowl.
The radius of each hemispherical bowl is half of the radius of the spherical ball:

\[ r_{hemisphere} = \frac{21}{2} = 10.5 \text{ cm} \]

### Step 5: Find the volume of one hemispherical bowl.
The volume \( V_h \) of a hemisphere is given by:

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

Substituting \( r = 10.5 \):

\[ V_h = \frac{2}{3} \pi (10.5)^3 \]

### Step 6: Calculate \( (10.5)^3 \).
Calculating \( 10.5^3 \):

\[ 10.5^3 = 10.5 \times 10.5 \times 10.5 = 110.25 \times 10.5 = 1157.625 \]

### Step 7: Substitute back into the volume formula for the hemisphere.
Now substituting \( 10.5^3 \) back into the volume formula:

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

### Step 8: Set the volume of the sphere equal to the total volume of the hemispherical bowls.
Let \( n \) be the number of hemispherical bowls. The total volume of \( n \) hemispherical bowls is:

\[ n \cdot V_h = n \cdot \frac{2315.25}{3} \pi \]

Setting the volumes equal:

\[ \frac{37044}{3} \pi = n \cdot \frac{2315.25}{3} \pi \]

### Step 9: Cancel out \( \frac{\pi}{3} \) from both sides.
This simplifies to:

\[ 37044 = n \cdot 2315.25 \]

### Step 10: Solve for \( n \).
Now, solving for \( n \):

\[ n = \frac{37044}{2315.25} \approx 16 \]

### Step 11: Find the total surface area of all hemispherical bowls.
The total surface area \( A \) of one hemispherical bowl is given by:

\[ A = 3 \pi r^2 \]

Substituting \( r = 10.5 \):

\[ A = 3 \pi (10.5)^2 \]

### Step 12: Calculate \( (10.5)^2 \).
Calculating \( (10.5)^2 \):

\[ (10.5)^2 = 110.25 \]

### Step 13: Substitute back into the surface area formula.
Now substituting \( (10.5)^2 \) back into the surface area formula:

\[ A = 3 \pi (110.25) = 330.75 \pi \]

### Step 14: Find the total surface area for \( n \) hemispherical bowls.
Now, the total surface area for \( n \) hemispherical bowls is:

\[ \text{Total Surface Area} = n \cdot A = 16 \cdot 330.75 \pi \]

### Step 15: Calculate the total surface area.
Calculating the total surface area:

\[ \text{Total Surface Area} = 16 \cdot 330.75 \cdot \frac{22}{7} \approx 16632 \text{ cm}^2 \]

### Final Answer:
Thus, the total surface area of all hemispherical bowls is:

\[ \text{Total Surface Area} = 16632 \text{ cm}^2 \]