The radius of a circular cylinder is the same as that of a sphere. Their volumes are equal. The height of the cylinder is ---

Options

A. 4/3 times its radius
B. 2/3 times its radius
C. Equal to its radius
D. Equal to its diameter

Answer

To find the height of the cylinder when the radius of the cylinder is the same as that of a sphere and their volumes are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formulas for the volumes:**
- The volume of a sphere (V_sphere) is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
- The volume of a cylinder (V_cylinder) is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]

2. **Set the volumes equal to each other:**
Since the volumes are equal, we can set the two equations equal:
\[ \frac{4}{3} \pi r^3 = \pi r^2 h \]

3. **Cancel out common terms:**
We can divide both sides of the equation by \(\pi\) (as long as \(\pi \neq 0\)) and simplify:
\[ \frac{4}{3} r^3 = r^2 h \]

4. **Isolate h:**
To find the height \(h\), we can rearrange the equation:
\[ h = \frac{4}{3} r^3 \cdot \frac{1}{r^2} \]
This simplifies to:
\[ h = \frac{4}{3} r \]

5. **Final answer:**
Therefore, the height of the cylinder is:
\[ h = \frac{4}{3} r \]