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 :
radius of the vessel.

Answer

To find the radius of the cylindrical vessel, we can follow these steps:

### Step 1: Understand the volume of the cylinder
We know that the volume \( V \) 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.

### Step 2: Set up the equations based on the given volumes
From the problem, we have two volumes:
1. The total volume of the cylinder when filled completely is \( 3080 \, \text{cm}^3 \).
2. The volume when filled up to 5 cm below the top is \( 2310 \, \text{cm}^3 \).

We can set up the first equation:
\[ \pi r^2 h = 3080 \tag{1} \]

For the second condition, if the height of the cylinder is \( h \), then the height of the water when filled up to 5 cm below the top is \( h - 5 \). Thus, we can set up the second equation:
\[ \pi r^2 (h - 5) = 2310 \tag{2} \]

### Step 3: Simplify the second equation
Expanding equation (2):
\[ \pi r^2 h - 5\pi r^2 = 2310 \]
We can substitute \( \pi r^2 h \) from equation (1):
\[ 3080 - 5\pi r^2 = 2310 \]

### Step 4: Solve for \( \pi r^2 \)
Rearranging the equation gives:
\[ 3080 - 2310 = 5\pi r^2 \]
\[ 770 = 5\pi r^2 \]
Now, divide both sides by \( 5 \):
\[ \pi r^2 = \frac{770}{5} = 154 \tag{3} \]

### Step 5: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 154 \]

### Step 6: Solve for \( r^2 \)
Multiplying both sides by \( 7 \):
\[ 22 r^2 = 154 \times 7 \]
Calculating \( 154 \times 7 \):
\[ 154 \times 7 = 1078 \]
Thus,
\[ 22 r^2 = 1078 \]
Now, divide by \( 22 \):
\[ r^2 = \frac{1078}{22} = 49 \]

### Step 7: Find \( r \)
Taking the square root of both sides:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Final Answer
The radius of the cylindrical vessel is \( 7 \, \text{cm} \).

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