Threee metallic cones of radius 2cm and height 9cm are melted and recast in to a solid sphere. Fuind the radius of the sphere.

Answer

To find the radius of the sphere formed by melting three metallic cones, we can follow these steps:

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

Given:
- Radius \( r = 2 \) cm
- Height \( h = 9 \) cm

Substituting the values:
\[ V = \frac{1}{3} \pi (2)^2 (9) \]
\[ V = \frac{1}{3} \pi (4)(9) \]
\[ V = \frac{36}{3} \pi \]
\[ V = 12\pi \text{ cm}^3 \]

### Step 2: Calculate the total volume of three cones.
Since there are three identical cones, the total volume \( V_{total} \) is:
\[ V_{total} = 3 \times V \]
\[ V_{total} = 3 \times 12\pi \]
\[ V_{total} = 36\pi \text{ cm}^3 \]

### Step 3: Set the total volume equal to 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.

Setting the total volume of the cones equal to the volume of the sphere:
\[ 36\pi = \frac{4}{3} \pi R^3 \]

### Step 4: Simplify the equation.
We can cancel \( \pi \) from both sides:
\[ 36 = \frac{4}{3} R^3 \]

### Step 5: Solve for \( R^3 \).
To eliminate the fraction, multiply both sides by \( 3 \):
\[ 108 = 4R^3 \]
Now, divide both sides by \( 4 \):
\[ R^3 = 27 \]

### Step 6: Find \( R \).
Taking the cube root of both sides:
\[ R = \sqrt[3]{27} \]
\[ R = 3 \text{ cm} \]

### Conclusion
The radius of the sphere is \( 3 \) cm.

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