A cone of height 4 cm is melted and recast into a sphere of diameter 8 cm .Find the radius of the base of the cone.

Answer

To find the radius of the base of the cone that is melted and recast into a sphere, we can follow these steps:

### Step 1: Understand the relationship between the volumes of the cone and the sphere.
Since the cone is melted and recast into a sphere, their volumes will be equal. We can denote the volume of the cone as \( V_1 \) and the volume of the sphere as \( V_2 \).

### Step 2: Write the formulas for the volumes.
The volume of a cone is given by the formula:
\[ V_1 = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

The volume of a sphere is given by the formula:
\[ V_2 = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the sphere.

### Step 3: Substitute the known values.
From the problem, we know:
- The height of the cone \( h = 4 \) cm.
- The diameter of the sphere is \( 8 \) cm, so the radius \( R = \frac{8}{2} = 4 \) cm.

### Step 4: Set the volumes equal to each other.
Since \( V_1 = V_2 \), we can write:
\[ \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi R^3 \]

### Step 5: Cancel out common factors.
We can cancel \( \pi \) and \( \frac{1}{3} \) from both sides:
\[ r^2 h = 4 R^3 \]

### Step 6: Substitute the known values into the equation.
Substituting \( h = 4 \) cm and \( R = 4 \) cm into the equation gives:
\[ r^2 \cdot 4 = 4 \cdot (4^3) \]

### Step 7: Calculate \( R^3 \).
Calculating \( 4^3 \):
\[ 4^3 = 64 \]
So, we have:
\[ r^2 \cdot 4 = 4 \cdot 64 \]
This simplifies to:
\[ r^2 \cdot 4 = 256 \]

### Step 8: Solve for \( r^2 \).
Dividing both sides by 4:
\[ r^2 = \frac{256}{4} = 64 \]

### Step 9: Find \( r \) by taking the square root.
Taking the square root of both sides gives:
\[ r = \sqrt{64} = 8 \text{ cm} \]

### Final Answer:
The radius of the base of the cone is \( 8 \) cm.

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