The total surface area of a cyclinder of diameter 10 cm is ` 330 cm ^(2)` Find the height of the cyclinder.

Answer

To find the height of the cylinder given its total surface area and diameter, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Total Surface Area of a Cylinder**:
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r (r + h) \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

2. **Identify Given Values**:
- Diameter of the cylinder = 10 cm
- Total Surface Area = 330 cm²

3. **Calculate the Radius**:
The radius \( r \) is half of the diameter:
\[ r = \frac{\text{Diameter}}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]

4. **Substitute Known Values into the TSA Formula**:
Substitute \( r = 5 \) cm and TSA = 330 cm² into the formula:
\[ 330 = 2\pi (5)(5 + h) \]

5. **Simplify the Equation**:
First, calculate \( 2\pi(5) \):
\[ 2\pi(5) = 10\pi \]
So the equation becomes:
\[ 330 = 10\pi(5 + h) \]

6. **Solve for \( 5 + h \)**:
Divide both sides by \( 10\pi \):
\[ 5 + h = \frac{330}{10\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ 10\pi \approx 10 \times \frac{22}{7} = \frac{220}{7} \]
Now substitute this back into the equation:
\[ 5 + h = \frac{330 \times 7}{220} = \frac{2310}{220} = 10.5 \]

7. **Isolate \( h \)**:
Now, subtract 5 from both sides:
\[ h = 10.5 - 5 = 5.5 \text{ cm} \]

### Final Answer:
The height of the cylinder is \( 5.5 \) cm.

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