3080 `cm^(3)` of water is required to fill a cylindrical vessel completely and 2310 `cm ^(3)` of water is required to fill it upto 5 cm below the top. Find :
height of the vessel.

Answer

To solve the problem, we need to find the height of the cylindrical vessel using the given volumes of water required to fill it.

### Step-by-Step Solution:

1. **Understand the Problem:**
We know that the volume of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

2. **Set Up the Equations:**
- The volume of the cylinder when completely filled is given as 3080 cm³:
\[ V_{full} = \pi r^2 h = 3080 \quad (1) \]
- The volume of the cylinder when filled up to 5 cm below the top is given as 2310 cm³:
\[ V_{5cm\ below\ top} = \pi r^2 (h - 5) = 2310 \quad (2) \]

3. **Divide the Two Equations:**
To eliminate \( \pi r^2 \), we can divide equation (1) by equation (2):
\[ \frac{\pi r^2 h}{\pi r^2 (h - 5)} = \frac{3080}{2310} \]
This simplifies to:
\[ \frac{h}{h - 5} = \frac{3080}{2310} \]

4. **Simplify the Right Side:**
Calculate \( \frac{3080}{2310} \):
\[ \frac{3080 \div 770}{2310 \div 770} = \frac{4}{3} \]
So, we have:
\[ \frac{h}{h - 5} = \frac{4}{3} \]

5. **Cross Multiply:**
Cross multiplying gives us:
\[ 3h = 4(h - 5) \]

6. **Expand and Rearrange:**
Expanding the right side:
\[ 3h = 4h - 20 \]
Rearranging gives:
\[ 4h - 3h = 20 \quad \Rightarrow \quad h = 20 \]

7. **Conclusion:**
The height of the cylindrical vessel is:
\[ h = 20 \text{ cm} \]