The height and base radius of a metallic cone are 216 cm and 16 cm respectively.It melted and recast into a sphere .Find the surface area of the spere.

Answer

To solve the problem step by step, we will find the surface area of the sphere formed by melting a cone with given dimensions.

### Step 1: Find the Volume of the Cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the base radius and \( h \) is the height of the cone.

Given:
- Base radius \( r = 16 \) cm
- Height \( h = 216 \) cm

Substituting the values into the volume formula:
\[ V = \frac{1}{3} \pi (16)^2 (216) \]

### Step 2: Calculate \( (16)^2 \)
Calculating the square of the radius:
\[ (16)^2 = 256 \]

### Step 3: Substitute and Simplify the Volume
Now substitute \( 256 \) into the volume formula:
\[ V = \frac{1}{3} \pi (256)(216) \]

### Step 4: Calculate \( 256 \times 216 \)
Calculating the product:
\[ 256 \times 216 = 55296 \]

### Step 5: Substitute Back to Find Volume
Now substitute back into the volume formula:
\[ V = \frac{1}{3} \pi (55296) \]
\[ V = 18432 \pi \text{ cm}^3 \]

### Step 6: Set Volume of Cone Equal to Volume of Sphere
Since the cone is melted and recast into a sphere, the volume of the sphere \( V_s \) is equal to the volume of the cone:
\[ V_s = \frac{4}{3} \pi R^3 \]
Setting the volumes equal:
\[ 18432 \pi = \frac{4}{3} \pi R^3 \]

### Step 7: Cancel \( \pi \) and Solve for \( R^3 \)
Cancelling \( \pi \) from both sides:
\[ 18432 = \frac{4}{3} R^3 \]

### Step 8: Multiply Both Sides by \( \frac{3}{4} \)
To isolate \( R^3 \):
\[ R^3 = 18432 \times \frac{3}{4} \]
\[ R^3 = 13824 \]

### Step 9: Calculate \( R \)
Now take the cube root of both sides to find \( R \):
\[ R = \sqrt[3]{13824} = 24 \text{ cm} \]

### Step 10: Find the Surface Area of the Sphere
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi R^2 \]
Substituting \( R = 24 \) cm:
\[ A = 4 \pi (24)^2 \]

### Step 11: Calculate \( (24)^2 \)
Calculating the square of the radius:
\[ (24)^2 = 576 \]

### Step 12: Substitute Back to Find Surface Area
Now substitute back into the surface area formula:
\[ A = 4 \pi (576) \]
\[ A = 2304 \pi \text{ cm}^2 \]

### Final Answer
The surface area of the sphere is:
\[ \boxed{2304 \pi \text{ cm}^2} \]