A right circular metal cone (solid) is 8 cm high and the radius is 2 cm. It is melted and recast into a sphere. What is the radius of the sphere?

Options

A. 2cm
B. 3cm
C. 4cm
D. 5cm

Answer

To find the radius of the sphere formed by melting a right circular cone, we will 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 and \( h \) is the height of the cone.

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

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

### 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 volume of the cone is equal to the volume of the sphere:
\[ \frac{32}{3} \pi = \frac{4}{3} \pi R^3 \]

### 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 the Radius \( R \)
To find \( R \), we take the cube root of both sides:
\[ R = \sqrt[3]{8} = 2 \text{ cm} \]

### Conclusion
The radius of the sphere is \( 2 \) cm.

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