The volume of a right circular cylinder with the height equal to the radius is `25 (1)/(7) cm^(3)`. Find the height of the cylinder. (Use `pi = (22)/(7)`).

Answer

To find the height of the right circular cylinder whose volume is given as \( 25 \frac{1}{7} \, \text{cm}^3 \) and where the height is equal to the radius, we can follow these steps:

### Step-by-Step Solution:

1. **Convert the mixed fraction to an improper fraction**:
\[ 25 \frac{1}{7} = \frac{25 \times 7 + 1}{7} = \frac{175 + 1}{7} = \frac{176}{7} \, \text{cm}^3 \]

2. **Use the formula for the volume of a cylinder**:
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Since the height \( h \) is equal to the radius \( r \), we can write:
\[ h = r \]
Therefore, substituting \( h \) with \( r \):
\[ V = \pi r^2 r = \pi r^3 \]

3. **Substitute the values into the volume formula**:
Given \( V = \frac{176}{7} \) and \( \pi = \frac{22}{7} \):
\[ \frac{176}{7} = \frac{22}{7} r^3 \]

4. **Eliminate the fraction by multiplying both sides by 7**:
\[ 176 = 22 r^3 \]

5. **Solve for \( r^3 \)**:
Divide both sides by 22:
\[ r^3 = \frac{176}{22} \]
Simplifying \( \frac{176}{22} \):
\[ r^3 = 8 \]

6. **Find the cube root of \( r^3 \)**:
\[ r = \sqrt[3]{8} = 2 \, \text{cm} \]

7. **Since height \( h \) is equal to radius \( r \)**:
\[ h = r = 2 \, \text{cm} \]

### Final Answer:
The height of the cylinder is \( 2 \, \text{cm} \).

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