The curved surface area and the diameter of a right circular cylinder are 132 `cm^(2)` and 7 cm respectively. Find its height (in cm).

Options

A. 7
B. 8
C. 9
D. 6

Answer

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

### Step 1: Identify the given values
- Curved Surface Area (CSA) = 132 cm²
- Diameter = 7 cm

### Step 2: Calculate the radius
The radius (r) of the cylinder can be calculated using the formula:
\[ r = \frac{\text{Diameter}}{2} \]
Substituting the given diameter:
\[ r = \frac{7 \text{ cm}}{2} = 3.5 \text{ cm} \]

### Step 3: Use the formula for the curved surface area of a cylinder
The formula for the curved surface area (CSA) of a cylinder is:
\[ \text{CSA} = 2 \pi r h \]
Where:
- \( \pi \) is approximately \( \frac{22}{7} \) or \( 3.14 \)
- \( h \) is the height of the cylinder

### Step 4: Substitute the known values into the formula
We can substitute the values we have into the CSA formula:
\[ 132 = 2 \times \frac{22}{7} \times 3.5 \times h \]

### Step 5: Simplify the equation
First, calculate \( 2 \times \frac{22}{7} \times 3.5 \):
- \( 2 \times \frac{22}{7} = \frac{44}{7} \)
- \( \frac{44}{7} \times 3.5 = \frac{44 \times 3.5}{7} = \frac{154}{7} \)

Now we can rewrite the equation:
\[ 132 = \frac{154}{7} \times h \]

### Step 6: Solve for height (h)
To isolate \( h \), multiply both sides by \( \frac{7}{154} \):
\[ h = 132 \times \frac{7}{154} \]
Calculating this gives:
\[ h = \frac{924}{154} = 6 \text{ cm} \]

### Conclusion
The height of the cylinder is \( 6 \text{ cm} \).

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