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

Answer

To solve the problem of finding the radius of a 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 of the cone.

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

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

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

### Step 3: Set the volume of the sphere equal to the total volume of the cones
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the sphere.

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

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

### Step 5: Solve for \( R^3 \)
Multiply both sides by \( \frac{3}{4} \):
\[ R^3 = 36 \times \frac{3}{4} = 27 \]

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

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

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