A sphere of diameter `12.6 cm` is melted and cast into a right circular cone of height `25.2 cm.` The radius of the base of the cone is

Options

A. `6.3 cm`
B. `2.1 cm`
C. `6 cm`
D. `4 cm`

Answer

To find the radius of the base of the cone formed by melting a sphere with a diameter of 12.6 cm, we can follow these steps:

### Step 1: Calculate the radius of the sphere
The diameter of the sphere is given as 12.6 cm. The radius (r) is half of the diameter.

\[ r = \frac{diameter}{2} = \frac{12.6 \, cm}{2} = 6.3 \, cm \]

### Step 2: Calculate the volume of the sphere
The formula for the volume (V) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

Substituting the radius we found:

\[ V = \frac{4}{3} \pi (6.3)^3 \]

Calculating \( (6.3)^3 \):

\[ (6.3)^3 = 6.3 \times 6.3 \times 6.3 = 250.047 \]

Now substituting back into the volume formula:

\[ V = \frac{4}{3} \pi (250.047) \approx \frac{1000.188}{3} \pi \approx 333.396 \pi \, cm^3 \]

### Step 3: Set the volume of the sphere equal to the volume of the cone
The volume 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 (25.2 cm). Thus, we have:

\[ \frac{4}{3} \pi (6.3)^3 = \frac{1}{3} \pi R^2 (25.2) \]

### Step 4: Cancel out π and simplify
We can cancel π from both sides:

\[ \frac{4}{3} (6.3)^3 = \frac{1}{3} R^2 (25.2) \]

Multiplying both sides by 3 to eliminate the fraction:

\[ 4 (6.3)^3 = R^2 (25.2) \]

### Step 5: Solve for \( R^2 \)
Substituting \( (6.3)^3 = 250.047 \):

\[ 4 \times 250.047 = R^2 \times 25.2 \]

Calculating the left side:

\[ 1000.188 = R^2 \times 25.2 \]

Now, divide both sides by 25.2:

\[ R^2 = \frac{1000.188}{25.2} \approx 39.69 \]

### Step 6: Calculate \( R \)
Taking the square root of both sides to find \( R \):

\[ R = \sqrt{39.69} \approx 6.29 \, cm \]

### Final Answer
The radius of the base of the cone is approximately **6.29 cm**.

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