A hollow hemispherical bowl is made of silver with its outer eadius 8 cm and inner radius 4 cm respectively. The bowl is melted to form a solid right circular cone of radius 8 cm. The height of the cone formed is

Options

A. a.7 cm
B. b.9 cm
C. c.12 cm
D. d.14 cm

Answer

To solve the problem step by step, we will calculate the height of the cone formed by melting the hollow hemispherical bowl.

### Step 1: Calculate the volume of the hollow hemispherical bowl.
The volume \( V \) of a hollow hemispherical bowl can be calculated using the formula:
\[ V = \frac{2}{3} \pi (R^3 - r^3) \]
where \( R \) is the outer radius and \( r \) is the inner radius.

Given:
- Outer radius \( R = 8 \) cm
- Inner radius \( r = 4 \) cm

Substituting the values into the formula:
\[ V = \frac{2}{3} \pi (8^3 - 4^3) \]

### Step 2: Calculate \( 8^3 \) and \( 4^3 \).
Calculating the cubes:
\[ 8^3 = 512 \quad \text{and} \quad 4^3 = 64 \]

### Step 3: Substitute the cubes back into the volume formula.
Now substituting these values back into the volume formula:
\[ V = \frac{2}{3} \pi (512 - 64) = \frac{2}{3} \pi (448) \]

### Step 4: Simplify the volume expression.
Now we simplify:
\[ V = \frac{2 \times 448}{3} \pi = \frac{896}{3} \pi \, \text{cm}^3 \]

### Step 5: Set the volume of the cone equal to the volume of the hemispherical bowl.
The volume \( V_c \) of a right circular cone is given by:
\[ V_c = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the cone and \( h \) is the height of the cone.

Given:
- Radius of the cone \( r = 8 \) cm

Substituting into the volume formula:
\[ V_c = \frac{1}{3} \pi (8^2) h = \frac{1}{3} \pi (64) h = \frac{64}{3} \pi h \]

### Step 6: Equate the volumes and solve for height \( h \).
Now, we set the volume of the cone equal to the volume of the hollow hemispherical bowl:
\[ \frac{896}{3} \pi = \frac{64}{3} \pi h \]

### Step 7: Cancel \( \pi \) and \( \frac{1}{3} \) from both sides.
Cancelling out \( \pi \) and \( \frac{1}{3} \):
\[ 896 = 64h \]

### Step 8: Solve for \( h \).
Now, divide both sides by 64:
\[ h = \frac{896}{64} = 14 \, \text{cm} \]

### Final Answer:
The height of the cone formed is \( h = 14 \) cm.

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