A hollow sphere of internal and external diameters 6 cm and 10 cm respectively is melted into a right circular cone of diameter 8 cm. The height of the cone is

Options

A. 22.5 cm
B. 23.5 cm
C. 24.5 cm
D. 25.5 cm

Answer

To solve the problem, we need to find the height of a right circular cone formed by melting a hollow sphere. Here are the steps to find the solution:

### Step 1: Find the radii of the hollow sphere
The hollow sphere has an external diameter of 10 cm and an internal diameter of 6 cm.

- External radius (R) = Diameter/2 = 10 cm / 2 = 5 cm
- Internal radius (r) = Diameter/2 = 6 cm / 2 = 3 cm

### Step 2: Calculate the volume of the hollow sphere
The volume \( V \) of a hollow sphere is given by the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
Substituting the values we found:
\[ V = \frac{4}{3} \pi (5^3 - 3^3) = \frac{4}{3} \pi (125 - 27) = \frac{4}{3} \pi (98) \]
\[ V = \frac{392}{3} \pi \text{ cm}^3 \]

### Step 3: Find the radius of the cone
The cone has a diameter of 8 cm, so the radius (r_cone) is:
\[ r_{cone} = \frac{8}{2} = 4 \text{ cm} \]

### Step 4: Write the formula for the volume of the cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r_{cone}^2 h \]
Substituting the radius of the cone:
\[ V = \frac{1}{3} \pi (4^2) h = \frac{1}{3} \pi (16) h = \frac{16}{3} \pi h \]

### Step 5: Set the volumes equal to each other
Since the volume of the hollow sphere is equal to the volume of the cone:
\[ \frac{392}{3} \pi = \frac{16}{3} \pi h \]

### Step 6: Cancel out \(\pi\) and solve for \(h\)
\[ \frac{392}{3} = \frac{16}{3} h \]
Multiplying both sides by 3:
\[ 392 = 16h \]
Now, divide both sides by 16:
\[ h = \frac{392}{16} = 24.5 \text{ cm} \]

### Conclusion
The height of the cone is \( 24.5 \) cm.

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