The area of the curved surface and the area of the base of a right circular cylinder are a sq cm and b gq cm respectively. The height of the cylinder is

Options

A. `(22 a )/(sqrt(pib))` cm
B. `(8sqrt(b))/(2 sqrt(pi))` cm
C. `(a)/(2sqrt(pib))` cm
D. `(asqrt(pi))/(2sqrt(b))` cm

Answer

To find the height of a right circular cylinder given the curved surface area (CSA) and the area of the base, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
- The formula for the area of the base of a cylinder is:
\[ \text{Area of base} = \pi r^2 \]

2. **Assign Variables**:
- Let the curved surface area be \( a \) cm².
- Let the area of the base be \( b \) cm².
- From the formulas, we can write:
\[ 2\pi rh = a \quad \text{(1)} \]
\[ \pi r^2 = b \quad \text{(2)} \]

3. **Express \( r \) in terms of \( b \)**:
- From equation (2), we can solve for \( r^2 \):
\[ r^2 = \frac{b}{\pi} \]
- Taking the square root gives us:
\[ r = \sqrt{\frac{b}{\pi}} \quad \text{(3)} \]

4. **Substitute \( r \) into equation (1)**:
- Substitute \( r \) from equation (3) into equation (1):
\[ 2\pi \left(\sqrt{\frac{b}{\pi}}\right)h = a \]
- Simplifying this gives:
\[ 2h\sqrt{b\pi} = a \]

5. **Solve for \( h \)**:
- Rearranging the equation to isolate \( h \):
\[ h = \frac{a}{2\sqrt{b\pi}} \quad \text{(4)} \]

6. **Final Result**:
- The height \( h \) of the cylinder is:
\[ h = \frac{a}{2\sqrt{b\pi}} \text{ cm} \]