A solid metallic sphere of radius 8.4 cm is melted and recast into a right circular cylinder of radius 12 cm. What is the height of the cylinder? (Your answer should be correct to one decimal place.)

Options

A. 7.0 cm
B. 6.0 cm
C. 6.5 cm
D. 5.5 cm

Answer

To solve the problem, we need to find the height of the cylinder formed by melting a solid metallic sphere. The volume of the sphere will be equal to the volume of the cylinder since the material is conserved during the melting and recasting process.

### 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. For our sphere, the radius \( r = 8.4 \) cm.
\[ V = \frac{4}{3} \pi (8.4)^3 \]

2. **Calculate \( (8.4)^3 \):**
\[ (8.4)^3 = 8.4 \times 8.4 \times 8.4 = 592.704 \text{ cm}^3 \]
So,
\[ V = \frac{4}{3} \pi (592.704) \]

3. **Calculate the Volume of the Sphere:**
\[ V = \frac{4}{3} \times \pi \times 592.704 \approx 2486.4 \text{ cm}^3 \text{ (using } \pi \approx 3.14\text{)} \]

4. **Calculate the Volume of the Cylinder:**
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height. For our cylinder, the radius \( r = 12 \) cm and we need to find the height \( h \).
\[ V = \pi (12)^2 h = \pi \times 144 h \]

5. **Set the Volumes Equal:**
Since the volume of the sphere is equal to the volume of the cylinder, we can set them equal to each other:
\[ \frac{4}{3} \pi (8.4)^3 = \pi (12)^2 h \]
Cancel \( \pi \) from both sides:
\[ \frac{4}{3} (8.4)^3 = 144 h \]

6. **Solve for \( h \):**
\[ h = \frac{\frac{4}{3} (8.4)^3}{144} \]
Substitute \( (8.4)^3 \approx 592.704 \):
\[ h = \frac{\frac{4}{3} \times 592.704}{144} \]
\[ h = \frac{790.272}{144} \approx 5.49 \text{ cm} \]

7. **Round to One Decimal Place:**
Rounding \( 5.49 \) to one decimal place gives:
\[ h \approx 5.5 \text{ cm} \]

### Final Answer:
The height of the cylinder is approximately **5.5 cm**.