A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere . What is the ratio of the height and radius of the cone so formed ?

Options

A. `4:3`
B. `2:3`
C. `3:4`
D. none of these

Answer

To solve the problem, we need to find the ratio of the height of the cone to the radius of the cone formed when a solid sphere is melted and recast into a right circular cone.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the radius of the sphere be \( r \).
- Since the radius of the cone is equal to the radius of the sphere, the radius of the cone \( R = r \).
- Let the height of the cone be \( h \).

2. **Volume of the Sphere**:
- The formula for the volume of a sphere is given by:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]

3. **Volume of the Cone**:
- The formula for the volume of a cone is given by:
\[ V_{cone} = \frac{1}{3} \pi R^2 h \]
- Since \( R = r \), we can substitute \( R \) in the volume formula of the cone:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

4. **Equate the Volumes**:
- Since the volume of the sphere is equal to the volume of the cone (as the sphere is melted and recast into the cone), we have:
\[ \frac{4}{3} \pi r^3 = \frac{1}{3} \pi r^2 h \]

5. **Cancel Common Terms**:
- We can cancel \( \frac{1}{3} \pi \) from both sides:
\[ 4r^3 = r^2 h \]

6. **Solve for Height \( h \)**:
- Divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = \frac{4r^3}{r^2} = 4r \]

7. **Find the Ratio of Height to Radius**:
- The ratio of the height of the cone \( h \) to the radius of the cone \( R \) is:
\[ \text{Ratio} = \frac{h}{R} = \frac{4r}{r} = 4 \]

### Final Answer:
The ratio of the height to the radius of the cone is \( 4:1 \).

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