The height and radius of base of a metallic cone are 27 cm and 16 cm respectively .It is melted and recast into a sphere .Find the radius and curved surface of the sphere.

Explanation

To solve the problem step by step, we need to find the radius and the curved surface area of a sphere formed by melting a metallic cone.

### Step 1: Identify the given values
- Height of the cone (h) = 27 cm
- Radius of the base of the cone (r) = 16 cm

### Step 2: Calculate the volume of the cone
The formula for the volume of a cone is given by:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Substituting the values of r and h:
\[ V_{cone} = \frac{1}{3} \pi (16)^2 (27) \]
Calculating \(16^2\):
\[ 16^2 = 256 \]
Now substituting back:
\[ V_{cone} = \frac{1}{3} \pi (256) (27) \]
Calculating \(256 \times 27\):
\[ 256 \times 27 = 6912 \]
Thus,
\[ V_{cone} = \frac{1}{3} \pi (6912) = 2304 \pi \, \text{cm}^3 \]

### Step 3: Set the volume of the cone equal to the volume of the sphere
When the cone is melted and recast into a sphere, the volume of the sphere (V_sphere) will be equal to the volume of the cone:
\[ V_{sphere} = \frac{4}{3} \pi R^3 \]
Setting the volumes equal:
\[ 2304 \pi = \frac{4}{3} \pi R^3 \]
We can cancel \(\pi\) from both sides:
\[ 2304 = \frac{4}{3} R^3 \]

### Step 4: Solve for \(R^3\)
Multiplying both sides by \(\frac{3}{4}\):
\[ R^3 = 2304 \times \frac{3}{4} \]
Calculating \(2304 \times \frac{3}{4}\):
\[ 2304 \times \frac{3}{4} = 1728 \]
Thus,
\[ R^3 = 1728 \]

### Step 5: Find the radius \(R\)
To find \(R\), we take the cube root of 1728:
\[ R = \sqrt[3]{1728} \]
Calculating the cube root:
\[ R = 12 \, \text{cm} \]

### Step 6: Calculate the curved surface area of the sphere
The formula for the curved surface area (CSA) of a sphere is:
\[ CSA = 4 \pi R^2 \]
Substituting the value of \(R\):
\[ CSA = 4 \pi (12)^2 \]
Calculating \(12^2\):
\[ 12^2 = 144 \]
Thus,
\[ CSA = 4 \pi (144) = 576 \pi \, \text{cm}^2 \]

### Final Answers
- Radius of the sphere \(R = 12 \, \text{cm}\)
- Curved surface area of the sphere \(CSA = 576 \pi \, \text{cm}^2\)