The volume of a cylinder of height 8 cm is 1232cm^(3). Find its curved surface area and the total surface area.

Answer

To solve the problem, we need to find the radius of the cylinder first using the given volume, and then calculate the curved surface area and total surface area.

### Step-by-Step Solution:

1. **Identify the given values:**
- Volume of the cylinder (V) = 1232 cm³
- Height of the cylinder (h) = 8 cm

2. **Use the formula for the volume of a cylinder:**
\[ V = \pi r^2 h \]
Here, \( \pi \) can be approximated as \( \frac{22}{7} \).

3. **Substitute the known values into the volume formula:**
\[ 1232 = \frac{22}{7} r^2 \times 8 \]

4. **Rearrange the equation to solve for \( r^2 \):**
\[ r^2 = \frac{1232 \times 7}{22 \times 8} \]

5. **Calculate the right side:**
- First, calculate \( 22 \times 8 = 176 \).
- Then, calculate \( 1232 \times 7 = 8624 \).
- Now, divide \( 8624 \) by \( 176 \):
\[ r^2 = \frac{8624}{176} = 49 \]

6. **Find the radius \( r \):**
\[ r = \sqrt{49} = 7 \text{ cm} \]

7. **Calculate the curved surface area (CSA) using the formula:**
\[ \text{CSA} = 2 \pi r h \]
Substitute the values:
\[ \text{CSA} = 2 \times \frac{22}{7} \times 7 \times 8 \]

8. **Simplify the calculation:**
- The \( 7 \) in the numerator and denominator cancels out:
\[ \text{CSA} = 2 \times 22 \times 8 = 352 \text{ cm}^2 \]

9. **Calculate the total surface area (TSA) using the formula:**
\[ \text{TSA} = \text{CSA} + 2 \pi r^2 \]
Substitute the values:
\[ \text{TSA} = 352 + 2 \times \frac{22}{7} \times 7^2 \]
- Calculate \( 7^2 = 49 \):
\[ \text{TSA} = 352 + 2 \times \frac{22}{7} \times 49 \]
- The \( 7 \) cancels out:
\[ \text{TSA} = 352 + 2 \times 22 \times 7 = 352 + 308 \]

10. **Final calculation for TSA:**
\[ \text{TSA} = 660 \text{ cm}^2 \]

### Final Answers:
- Curved Surface Area (CSA) = 352 cm²
- Total Surface Area (TSA) = 660 cm²