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

Options

A. a)`(2a)/(sqrt(pi b)) cm`
B. b)`(asqrt(b))/(2sqrt(pi)) cm`
C. c)`(a)/(2sqrt(pi b)) cm`
D. d)`(a sqrt(pi))/(2sqrt(b)) cm`

Answer

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

### Step 1: Understand the formulas
The formulas we need are:
- Curved Surface Area (CSA) of a cylinder: \( CSA = 2 \pi r h \)
- Area of the base of a cylinder: \( Area_{base} = \pi r^2 \)

### Step 2: Set up the equations
From the problem, we know:
- The curved surface area is given as \( a \) square cm, so we can write:
\[ 2 \pi r h = a \]
- The area of the base is given as \( b \) square cm, so we can write:
\[ \pi r^2 = b \]

### Step 3: Solve for \( r \)
From the area of the base equation, we can solve for \( r^2 \):
\[ r^2 = \frac{b}{\pi} \]
Taking the square root gives us:
\[ r = \sqrt{\frac{b}{\pi}} \]

### Step 4: Substitute \( r \) back into the CSA equation
Now, substitute \( r \) into the CSA equation:
\[ 2 \pi \left(\sqrt{\frac{b}{\pi}}\right) h = a \]
This simplifies to:
\[ 2 \sqrt{\pi} b^{1/2} h = a \]

### Step 5: Solve for \( h \)
Now, isolate \( h \):
\[ h = \frac{a}{2 \sqrt{\pi} \sqrt{b}} \]
This can be rewritten as:
\[ h = \frac{a}{2 \sqrt{\pi b}} \]

### Final Answer
Thus, the height of the cylinder \( h \) is:
\[ h = \frac{a}{2 \sqrt{\pi b}} \]

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