A sphere of radius 6 cm is melted and reacst in to a cone of height 6 cm .Find the radius of the cone.

Explanation

To solve the problem of finding the radius of a cone formed by melting a sphere of radius 6 cm, we will follow these steps:

### Step 1: Find the Volume of the Sphere
The formula for the volume of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

where \( r \) is the radius of the sphere. Here, the radius \( r = 6 \) cm.

Substituting the value of \( r \):

\[ V = \frac{4}{3} \pi (6)^3 \]

Calculating \( (6)^3 \):

\[ (6)^3 = 216 \]

Now substituting back into the volume formula:

\[ V = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \text{ cm}^3 \]

### Step 2: Find 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 cone and \( h \) is the height of the cone. Given that the height \( h = 6 \) cm, we can write:

\[ V = \frac{1}{3} \pi r^2 (6) = 2 \pi r^2 \]

### Step 3: Set the Volumes Equal
Since the sphere is melted and recast into the cone, the volumes of the sphere and cone are equal:

\[ 288 \pi = 2 \pi r^2 \]

### Step 4: Simplify the Equation
We can cancel \( \pi \) from both sides:

\[ 288 = 2 r^2 \]

Now, divide both sides by 2:

\[ r^2 = \frac{288}{2} = 144 \]

### Step 5: Solve for \( r \)
Taking the square root of both sides gives:

\[ r = \sqrt{144} = 12 \text{ cm} \]

### Conclusion
The radius of the cone is \( 12 \) cm.

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