The perimeter of the base of a right circular cylinder is 'a' unit. If the volume of the cylinder is V cubic unit, then the height of the cylinder is

Options

A. `(4a^(2)V)/(pi)` unit
B. `(4pi a^(2))/(V)` unit
C. `(pipa^(2)V)/(4)` unit
D. `(4 pi V)/(a^(2))` unit

Answer

To find the height of a right circular cylinder given the perimeter of its base and its volume, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Perimeter of the Base**:
The base of a right circular cylinder is a circle. The perimeter (circumference) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

2. **Set the Perimeter Equal to 'a'**:
According to the problem, the perimeter of the base is given as \( a \) units. Therefore, we can write:
\[ 2\pi r = a \]

3. **Solve for the Radius \( r \)**:
Rearranging the equation to solve for \( r \):
\[ r = \frac{a}{2\pi} \]

4. **Volume of the Cylinder**:
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( h \) is the height of the cylinder.

5. **Substitute the Value of \( r \)**:
Substitute \( r = \frac{a}{2\pi} \) into the volume formula:
\[ V = \pi \left(\frac{a}{2\pi}\right)^2 h \]

6. **Simplify the Volume Equation**:
Simplifying the equation:
\[ V = \pi \cdot \frac{a^2}{4\pi^2} \cdot h \]
\[ V = \frac{a^2}{4\pi} h \]

7. **Solve for Height \( h \)**:
Rearranging the equation to solve for \( h \):
\[ h = \frac{4\pi V}{a^2} \]

### Final Answer:
The height of the cylinder is:
\[ h = \frac{4\pi V}{a^2} \]