Explanation:

To find the height of the cone:

- Volume of the sphere:

Formula: \(\frac{4}{3} \pi r^3\).

Radius = 14 cm.

Volume = \(\frac{4}{3} \pi (14)^3 = \frac{4}{3} \pi \times 2744\).

- Volume of the cone:

Formula: \(\frac{1}{3} \pi r^2 h\).

Diameter = 14 cm, so radius = 7 cm.

Let the height be \(h\).

- Equate the volumes (since the metal is melted and recast):

\(\frac{4}{3} \pi \times 2744 = \frac{1}{3} \pi \times 7^2 \times h\).

- Solve for \(h\):

\(4 \times 2744 = 49 \times h\).

\(h = \frac{4 \times 2744}{49} = 224\) cm.

- Therefore, the height of the cone is \(\textbf{224 cm}\).

- Correct Answer: Option 4 – 224