If the volume and curved surface area of a cylinder are `269.5 cm^(3)` and ` 154 cm^(2)` respectively , what is the height of the cylinder ?

Options

A. 6
B. 3.5
C. 7
D. can't be determined

Answer

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

### Step-by-Step Solution:

1. **Identify the formulas**:
- The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
- The curved surface area \( A \) of a cylinder is given by the formula:
\[ A = 2 \pi r h \]

2. **Substitute the given values**:
- We know that the volume \( V = 269.5 \, \text{cm}^3 \) and the curved surface area \( A = 154 \, \text{cm}^2 \).

3. **Set up the equations**:
- From the volume equation:
\[ \pi r^2 h = 269.5 \quad \text{(1)} \]
- From the curved surface area equation:
\[ 2 \pi r h = 154 \quad \text{(2)} \]

4. **Rearranging equation (2) to find \( h \)**:
- We can express \( h \) in terms of \( r \):
\[ h = \frac{154}{2 \pi r} \quad \text{(3)} \]

5. **Substituting equation (3) into equation (1)**:
- Substitute \( h \) from equation (3) into equation (1):
\[ \pi r^2 \left(\frac{154}{2 \pi r}\right) = 269.5 \]
- Simplifying this gives:
\[ \frac{154 r}{2} = 269.5 \]
- Further simplifying:
\[ 154 r = 539 \quad \text{(multiplying both sides by 2)} \]
- Thus:
\[ r = \frac{539}{154} \approx 3.5 \, \text{cm} \]

6. **Finding the height using the radius**:
- Now substitute \( r = 3.5 \, \text{cm} \) back into equation (3) to find \( h \):
\[ h = \frac{154}{2 \pi (3.5)} \]
- Using \( \pi \approx \frac{22}{7} \):
\[ h = \frac{154}{2 \times \frac{22}{7} \times 3.5} \]
- Simplifying the denominator:
\[ 2 \times \frac{22}{7} \times 3.5 = \frac{154}{7} \]
- Thus:
\[ h = \frac{154}{\frac{154}{7}} = 7 \, \text{cm} \]

### Final Answer:
The height of the cylinder is \( 7 \, \text{cm} \).

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