A right circular cone is 8 cm high and radius of its base is 2 cm .The cone is metted and recaset in to a sphere. Determine the diameter of the sphere.

Answer

To solve the problem step by step, we will calculate the diameter of the sphere formed by melting the cone.

### Step 1: Find the volume of the cone
The formula for the volume \( V \) of a right circular 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 of the cone.

Given:
- Height \( h = 8 \) cm
- Radius \( r = 2 \) cm

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (2)^2 (8) \]
Calculating this:
\[ V = \frac{1}{3} \pi (4)(8) = \frac{32}{3} \pi \text{ cm}^3 \]

### Step 2: Set the volume of the sphere equal to the volume of the cone
When the cone is melted and recast into a sphere, the volume of the sphere \( V_s \) will be equal to the volume of the cone \( V_c \).

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

Setting the volumes equal:
\[ \frac{32}{3} \pi = \frac{4}{3} \pi R^3 \]

### Step 3: Cancel out common terms
We can cancel \( \pi \) and \( \frac{1}{3} \) from both sides:
\[ 32 = 4 R^3 \]

### Step 4: Solve for \( R^3 \)
Dividing both sides by 4:
\[ R^3 = \frac{32}{4} = 8 \]

### Step 5: Find \( R \)
Taking the cube root of both sides:
\[ R = 8^{1/3} = 2 \text{ cm} \]

### Step 6: Calculate the diameter of the sphere
The diameter \( D \) of the sphere is given by:
\[ D = 2R \]
Substituting the value of \( R \):
\[ D = 2 \times 2 = 4 \text{ cm} \]

### Final Answer
The diameter of the sphere is \( 4 \) cm.

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