Twelve solid hemispheres of the same size are melted and recast in a right circular cylinder of diameter 7 cm and height 28 cm. What is the radius of the hemispheres?

Options

A. 4.5 cm
B. 3.5 cm
C. 3 cm
D. 3.8 cm

Answer

To solve the problem, we need to find the radius of the hemispheres when 12 solid hemispheres are melted and recast into a right circular cylinder.

### Step-by-Step Solution:

1. **Understand the Volume Relationship**:
The volume of the 12 hemispheres will be equal to the volume of the cylinder since the material is conserved during the melting and recasting process.

2. **Volume of a Hemisphere**:
The formula for the volume of a single hemisphere is:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
Therefore, the volume of 12 hemispheres is:
\[ V_{\text{12 hemispheres}} = 12 \times \frac{2}{3} \pi r^3 = 8 \pi r^3 \]

3. **Volume of the Cylinder**:
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi R^2 h \]
where \( R \) is the radius and \( h \) is the height. Given the diameter of the cylinder is 7 cm, the radius \( R \) is:
\[ R = \frac{7}{2} = 3.5 \text{ cm} \]
The height \( h \) of the cylinder is 28 cm. Thus, the volume of the cylinder is:
\[ V_{\text{cylinder}} = \pi (3.5)^2 (28) \]

4. **Calculate the Volume of the Cylinder**:
First, calculate \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now, substitute this into the volume formula:
\[ V_{\text{cylinder}} = \pi \times 12.25 \times 28 = 343 \pi \]

5. **Set the Volumes Equal**:
Now, we set the volume of the 12 hemispheres equal to the volume of the cylinder:
\[ 8 \pi r^3 = 343 \pi \]
We can cancel \( \pi \) from both sides:
\[ 8 r^3 = 343 \]

6. **Solve for \( r^3 \)**:
Divide both sides by 8:
\[ r^3 = \frac{343}{8} \]

7. **Calculate \( r \)**:
Now, take the cube root of both sides:
\[ r = \sqrt[3]{\frac{343}{8}} = \frac{\sqrt[3]{343}}{\sqrt[3]{8}} = \frac{7}{2} = 3.5 \text{ cm} \]

### Final Answer:
The radius of the hemispheres is \( 3.5 \) cm.