A cone id 8.4 cm height and the radius of its base is 2.1 cm. It is melted and recast into a sphere. The radius of the sphere is .

Options

A. 4.2 cm
B. 2.1 cm
C. 2.4 cm
D. 1.6 cm

Explanation

To find the radius of the sphere formed by melting the cone, we need to 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 of the base and \( h \) is the height of the cone.

Given:
- Height \( h = 8.4 \) cm
- Radius \( r = 2.1 \) cm

Substituting the values into the formula:

\[ V = \frac{1}{3} \pi (2.1)^2 (8.4) \]

### Step 2: Calculate \( (2.1)^2 \).
Calculating \( (2.1)^2 \):

\[ (2.1)^2 = 4.41 \]

### Step 3: Substitute back into the volume formula.
Now substituting \( (2.1)^2 \) back into the volume formula:

\[ V = \frac{1}{3} \pi (4.41) (8.4) \]

### Step 4: Calculate \( 4.41 \times 8.4 \).
Calculating \( 4.41 \times 8.4 \):

\[ 4.41 \times 8.4 = 37.044 \]

### Step 5: Substitute this value into the volume formula.
Now substituting this value back into the volume formula:

\[ V = \frac{1}{3} \pi (37.044) \]

### Step 6: Calculate the volume.
Now, calculating the volume:

\[ V = \frac{37.044 \pi}{3} \approx 12.348 \pi \, \text{cm}^3 \]

### Step 7: Set the volume of the sphere equal to the volume of the cone.
The volume of a sphere is given by:

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

where \( R \) is the radius of the sphere. Setting the volume of the cone equal to the volume of the sphere:

\[ \frac{37.044 \pi}{3} = \frac{4}{3} \pi R^3 \]

### Step 8: Cancel \( \pi \) and multiply by 3.
Cancelling \( \pi \) from both sides and multiplying by 3:

\[ 37.044 = 4 R^3 \]

### Step 9: Solve for \( R^3 \).
Dividing both sides by 4:

\[ R^3 = \frac{37.044}{4} \approx 9.261 \]

### Step 10: Take the cube root to find \( R \).
Now, take the cube root of both sides to find \( R \):

\[ R \approx \sqrt[3]{9.261} \approx 2.1 \, \text{cm} \]

### Final Answer:
The radius of the sphere is approximately \( 2.1 \) cm.

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