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^2v)/(pi)` unit
B. `(4pia^2)/(v)` unit
C. `(pia^2v)/(4)` unit
D. `(4piV)/(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 1: Understand the given information
We know:
- The perimeter (circumference) of the base of the cylinder is \( a \) units.
- The volume of the cylinder is \( V \) cubic units.

### Step 2: Relate the perimeter to the radius
The formula for the perimeter (circumference) of the base of a cylinder is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the base. Since the perimeter is given as \( a \):
\[ 2\pi r = a \]
From this, we can solve for \( r \):
\[ r = \frac{a}{2\pi} \]

### Step 3: Write the formula for the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( h \) is the height of the cylinder.

### Step 4: Substitute the expression for \( r \) into the volume formula
Substituting \( r = \frac{a}{2\pi} \) into the volume formula:
\[ V = \pi \left(\frac{a}{2\pi}\right)^2 h \]
Calculating \( \left(\frac{a}{2\pi}\right)^2 \):
\[ \left(\frac{a}{2\pi}\right)^2 = \frac{a^2}{4\pi^2} \]
Thus, substituting back into the volume formula gives:
\[ V = \pi \cdot \frac{a^2}{4\pi^2} \cdot h \]
This simplifies to:
\[ V = \frac{a^2 h}{4\pi} \]

### Step 5: Solve for the height \( h \)
To find \( h \), we rearrange the equation:
\[ h = \frac{4\pi V}{a^2} \]

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