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

Answer

To find the radius of the base of the cone formed from a metallic sphere of radius 6 cm, we will follow these steps:

### Step 1: Calculate the volume of the metallic sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Here, \( r = 6 \) cm.

Substituting the value of \( r \):
\[ V = \frac{4}{3} \pi (6)^3 \]
Calculating \( 6^3 \):
\[ 6^3 = 216 \]
So,
\[ V = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \, \text{cm}^3 \]

### Step 2: Set the volume of the cone equal to the volume of the sphere
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 of the cone and \( h \) is the height of the cone. Here, \( h = 24 \) cm.

Setting the volume of the cone equal to the volume of the sphere:
\[ \frac{1}{3} \pi r^2 (24) = 288 \pi \]

### Step 3: Simplify the equation
First, we can cancel \( \pi \) from both sides:
\[ \frac{1}{3} r^2 (24) = 288 \]
Now, simplify the left side:
\[ 8 r^2 = 288 \]

### Step 4: Solve for \( r^2 \)
Now, divide both sides by 8:
\[ r^2 = \frac{288}{8} = 36 \]

### Step 5: Find the radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{36} = 6 \, \text{cm} \]

### Final Answer
The radius of the base of the cone is \( 6 \, \text{cm} \).

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