A sphere of diameter 15.6 cm is melted and cast into a right circular cone of height 31.2 cm. Find the diameter of the base of the cone.

Explanation

To solve the problem, we need to find the diameter of the base of a cone that is formed by melting a sphere. The volume of the sphere will be equal to the volume of the cone.

### Step-by-Step Solution:

**Step 1: Calculate the radius of the sphere.**
- The diameter of the sphere is given as 15.6 cm.
- The radius \( r \) of the sphere can be calculated using the formula:
\[ r = \frac{\text{diameter}}{2} = \frac{15.6 \, \text{cm}}{2} = 7.8 \, \text{cm} \]

**Step 2: Calculate the volume of the sphere.**
- The formula for the volume \( V \) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
- Substituting the radius we found:
\[ V = \frac{4}{3} \pi (7.8)^3 \]
- Calculating \( (7.8)^3 \):
\[ (7.8)^3 = 456.5332 \, \text{cm}^3 \quad (\text{approximately}) \]
- Now substituting back into the volume formula:
\[ V \approx \frac{4}{3} \pi (456.5332) \approx 1910.633 \, \text{cm}^3 \]

**Step 3: Set the volume of the cone equal to the volume of the sphere.**
- The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
- Here, \( h \) (height of the cone) is given as 31.2 cm. We need to find the radius \( R \) of the base of the cone.
- Setting the volumes equal:
\[ \frac{4}{3} \pi (7.8)^3 = \frac{1}{3} \pi R^2 (31.2) \]

**Step 4: Simplify the equation.**
- Cancel \( \pi \) from both sides:
\[ \frac{4}{3} (7.8)^3 = \frac{1}{3} R^2 (31.2) \]
- Multiply both sides by 3:
\[ 4 (7.8)^3 = R^2 (31.2) \]
- Now substitute \( (7.8)^3 \):
\[ 4 (456.5332) = R^2 (31.2) \]
- Calculate \( 4 \times 456.5332 \):
\[ 1826.133 = R^2 (31.2) \]

**Step 5: Solve for \( R^2 \).**
- Divide both sides by 31.2:
\[ R^2 = \frac{1826.133}{31.2} \approx 58.5 \]

**Step 6: Calculate \( R \).**
- Taking the square root:
\[ R = \sqrt{58.5} \approx 7.65 \, \text{cm} \]

**Step 7: Find the diameter of the cone.**
- The diameter \( D \) of the cone is given by:
\[ D = 2R = 2 \times 7.65 \approx 15.3 \, \text{cm} \]

### Final Answer:
The diameter of the base of the cone is approximately **15.3 cm**.