A cone is 8.4 cm heigh 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

Answer

To find the radius of the sphere formed by melting a cone, we will first calculate the volume of the cone and then set it equal to the volume of the sphere.

### Step-by-Step Solution:

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.

Here, \( r = 2.1 \) cm and \( h = 8.4 \) cm. Plugging in these values:
\[ V = \frac{1}{3} \pi (2.1)^2 (8.4) \]

2. **Calculate \( (2.1)^2 \)**:
\[ (2.1)^2 = 4.41 \]

3. **Substitute back into the volume formula**:
\[ V = \frac{1}{3} \pi (4.41)(8.4) \]

4. **Calculate \( 4.41 \times 8.4 \)**:
\[ 4.41 \times 8.4 = 37.044 \]

5. **Now calculate the volume**:
\[ V = \frac{1}{3} \pi (37.044) \]
\[ V = \frac{37.044}{3} \pi = 12.348 \pi \text{ cm}^3 \]

6. **Volume of the Sphere**:
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Setting the volume of the cone equal to the volume of the sphere:
\[ 12.348 \pi = \frac{4}{3} \pi r^3 \]

7. **Cancel \( \pi \) from both sides**:
\[ 12.348 = \frac{4}{3} r^3 \]

8. **Multiply both sides by \( \frac{3}{4} \)** to isolate \( r^3 \):
\[ r^3 = 12.348 \times \frac{3}{4} \]
\[ r^3 = 9.261 \]

9. **Calculate the cube root of both sides to find \( r \)**:
\[ r = \sqrt[3]{9.261} \approx 2.1 \text{ cm} \]

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