A hollow sphere of internal and external diameters 4 cm and 8 cm, respectively is melted into a solid cone of base diameter 8 cm. The height of the cone is :

Question

Accepted Answer

To determine the height of the cone formed by melting the hollow sphere, we need to equate the volume of the hollow sphere to the volume of the cone.

**Step 1: Calculate the volume of the hollow sphere**

The volume of a sphere is given by the formula:

$$ V = \frac{4}{3} \pi r^3 $$

For a hollow sphere, we need to calculate the volume of the outer sphere and subtract the volume of the inner sphere.

- **Outer sphere radius (R):** The external diameter is 8 cm, so the radius is $ R = \frac{8}{2} = 4 $ cm.
- **Inner sphere radius (r):** The internal diameter is 4 cm, so the radius is $ r = \frac{4}{2} = 2 $ cm.

The volume of the hollow sphere is:

$$ V_{\text{hollow sphere}} = \frac{4}{3} \pi (R^3 - r^3) $$

$$ V_{\text{hollow sphere}} = \frac{4}{3} \pi (4^3 - 2^3) $$

$$ V_{\text{hollow sphere}} = \frac{4}{3} \pi (64 - 8) $$

$$ V_{\text{hollow sphere}} = \frac{4}{3} \pi \times 56 $$

$$ V_{\text{hollow sphere}} = \frac{224}{3} \pi $$

**Step 2: Calculate the volume of the cone**

The volume of a cone is given by the formula:

$$ V = \frac{1}{3} \pi r^2 h $$

Where $ r $ is the radius of the base and $ h $ is the height of the cone.

- **Base diameter of the cone:** 8 cm, so the radius $ r = \frac{8}{2} = 4 $ cm.

Equating the volume of the hollow sphere to the volume of the cone:

$$ \frac{224}{3} \pi = \frac{1}{3} \pi \times 4^2 \times h $$

$$ \frac{224}{3} \pi = \frac{1}{3} \pi \times 16 \times h $$

Cancel $\pi$ from both sides:

$$ \frac{224}{3} = \frac{16}{3} \times h $$

Multiply both sides by 3 to eliminate the fraction:

$$ 224 = 16h $$

Divide both sides by 16 to solve for $ h $:

$$ h = \frac{224}{16} $$

$$ h = 14 $$

Therefore, the height of the cone is 14 cm, which corresponds to Option 3.

**Explanation for Incorrect Options:**

- **Option 1 (11 cm):** This height would result in a smaller volume than the volume of the hollow sphere, thus not conserving the material.
- **Option 2 (12 cm):** Similar to Option 1, this height would not account for the entire volume of the hollow sphere.
- **Option 4 (16 cm):** This height would imply a larger volume than the available material from the hollow sphere, which is not possible.

Thus, Option 3 is the correct answer, as it accurately reflects the conservation of volume when the hollow sphere is melted into the cone.