A cone of metal of height 24 cm and radius of base 6 cm is melted and recast into a sphere. Find the radius of the sphere.

Answer

To find the radius of the sphere formed by melting a cone, we need to follow these steps:

### Step 1: Calculate the Volume of the 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 of the base and \( h \) is the height.

Given:
- Radius of the base of the cone, \( r = 6 \) cm
- Height of the cone, \( h = 24 \) cm

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (6)^2 (24) \]
Calculating \( (6)^2 \):
\[ (6)^2 = 36 \]
Now substituting back:
\[ V = \frac{1}{3} \pi (36)(24) \]
Calculating \( 36 \times 24 \):
\[ 36 \times 24 = 864 \]
So,
\[ V = \frac{1}{3} \pi (864) = 288 \pi \text{ cm}^3 \]

### Step 2: Set the Volume of the Sphere Equal to the Volume of the Cone
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the sphere.

Since the cone is melted and recast into a sphere, the volumes are equal:
\[ 288 \pi = \frac{4}{3} \pi R^3 \]

### Step 3: Cancel \( \pi \) from Both Sides
Dividing both sides by \( \pi \):
\[ 288 = \frac{4}{3} R^3 \]

### Step 4: Solve for \( R^3 \)
To eliminate the fraction, multiply both sides by \( \frac{3}{4} \):
\[ R^3 = 288 \times \frac{3}{4} \]
Calculating \( 288 \times \frac{3}{4} \):
\[ 288 \times \frac{3}{4} = 216 \]
So,
\[ R^3 = 216 \]

### Step 5: Find \( R \) by Taking the Cube Root
To find \( R \), take the cube root of both sides:
\[ R = \sqrt[3]{216} \]
Calculating the cube root:
\[ R = 6 \text{ cm} \]

### Final Answer
The radius of the sphere is \( 6 \) cm.

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