A solid metallic sphere of radius 6.3 cm is melted and recast into a right circular cone of height 25.2 cm. What is the ratio of the diameter of the base to the height of the cone ?

Options

A. a)`2:1`
B. b)`3:2`
C. c)`1:2`
D. d)`2:3`

Answer

To solve the problem, we need to find the ratio of the diameter of the base of a cone to its height after melting a solid metallic sphere into that cone.

### Step-by-Step Solution:

1. **Calculate the Volume of the 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. Given that the radius \( r = 6.3 \) cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (6.3)^3 \]

2. **Calculate \( (6.3)^3 \):**
First, we calculate \( 6.3^3 \):
\[ 6.3^3 = 6.3 \times 6.3 \times 6.3 = 250.047 \]
Now substitute this back into the volume formula:
\[ V = \frac{4}{3} \pi (250.047) \]

3. **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 of the base of the cone and \( h \) is the height of the cone. The height \( h \) is given as \( 25.2 \) cm.

4. **Set the Volumes Equal:**
Since the sphere is melted and recast into the cone, the volumes must be equal:
\[ \frac{4}{3} \pi (6.3)^3 = \frac{1}{3} \pi r^2 (25.2) \]
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} (6.3)^3 = \frac{1}{3} r^2 (25.2) \]

5. **Simplify the Equation:**
Multiply both sides by 3 to eliminate the fraction:
\[ 4 (6.3)^3 = r^2 (25.2) \]
Now, we can solve for \( r^2 \):
\[ r^2 = \frac{4 (6.3)^3}{25.2} \]

6. **Calculate \( r^2 \):**
Substitute \( (6.3)^3 = 250.047 \):
\[ r^2 = \frac{4 \times 250.047}{25.2} \]
Calculate \( r^2 \):
\[ r^2 = \frac{1000.188}{25.2} \approx 39.7 \]
So, \( r \approx \sqrt{39.7} \approx 6.3 \) cm.

7. **Calculate the Diameter of the Base:**
The diameter \( d \) of the base of the cone is given by:
\[ d = 2r \approx 2 \times 6.3 = 12.6 \text{ cm} \]

8. **Calculate the Ratio of the Diameter to the Height:**
The height \( h \) of the cone is \( 25.2 \) cm. Therefore, the ratio of the diameter of the base to the height is:
\[ \text{Ratio} = \frac{d}{h} = \frac{12.6}{25.2} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{1}{2} \]

### Final Answer:
The ratio of the diameter of the base to the height of the cone is \( 1:2 \).