The total surface area of a closed cylinder is 528 `cm^(2)`. If its radius is 7 cm, find its height.

Answer

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

### Step-by-Step Solution:

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

2. **Substitute the Given Values**:
We know from the problem that the total surface area (TSA) is 528 cm² and the radius \( r \) is 7 cm. Substitute these values into the formula:
\[ 528 = 2\pi (7)(7 + h) \]

3. **Calculate the Left Side**:
First, calculate \( 2\pi \) when \( \pi \) is approximated as \( \frac{22}{7} \):
\[ 2\pi = 2 \times \frac{22}{7} = \frac{44}{7} \]
Now substitute this back into the equation:
\[ 528 = \frac{44}{7} (7 + h) \]

4. **Clear the Fraction**:
To eliminate the fraction, multiply both sides of the equation by 7:
\[ 528 \times 7 = 44(7 + h) \]
Calculate \( 528 \times 7 \):
\[ 3696 = 44(7 + h) \]

5. **Divide Both Sides by 44**:
Now divide both sides by 44 to isolate \( 7 + h \):
\[ 7 + h = \frac{3696}{44} \]
Calculate \( \frac{3696}{44} \):
\[ 7 + h = 84 \]

6. **Solve for Height \( h \)**:
Now, solve for \( h \) by subtracting 7 from both sides:
\[ h = 84 - 7 \]
Therefore,
\[ h = 77 \text{ cm} \]

### Final Answer:
The height \( h \) of the cylinder is **77 cm**.