A cylindrical water tank of diameter 2.8 m and height 4.2 m is being fed by a pipe of diameter 7 cm through which water flows at the rate of 4 `m s^(-1).` Calculate, in minutes, the time it takes to fill the tank

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the volume of the cylindrical water tank.
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Diameter of the tank \( D = 2.8 \, \text{m} \)
- Radius \( r = \frac{D}{2} = \frac{2.8}{2} = 1.4 \, \text{m} \)
- Height \( h = 4.2 \, \text{m} \)

Now substituting the values:
\[ V = \pi (1.4)^2 (4.2) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V \approx \frac{22}{7} \times (1.4)^2 \times 4.2 \]
Calculating \( (1.4)^2 = 1.96 \):
\[ V \approx \frac{22}{7} \times 1.96 \times 4.2 \]
Calculating further:
\[ V \approx \frac{22 \times 1.96 \times 4.2}{7} \approx \frac{22 \times 8.232}{7} \approx \frac{181.104}{7} \approx 25.872 \, \text{m}^3 \]

### Step 2: Calculate the volume of water flowing through the pipe.
The volume of water flowing through a cylindrical pipe is also given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the pipe and \( h \) is the height of the water column in the pipe.

Given:
- Diameter of the pipe \( D_2 = 7 \, \text{cm} = 0.07 \, \text{m} \)
- Radius \( r_2 = \frac{D_2}{2} = \frac{0.07}{2} = 0.035 \, \text{m} \)

The flow rate of water is given as \( 4 \, \text{m/s} \). The volume flow rate \( Q \) can be calculated as:
\[ Q = A \cdot v \]
where \( A \) is the cross-sectional area of the pipe and \( v \) is the velocity of the water.

Calculating the area \( A \):
\[ A = \pi r_2^2 = \pi (0.035)^2 \approx \frac{22}{7} \times (0.035)^2 \]
Calculating \( (0.035)^2 = 0.001225 \):
\[ A \approx \frac{22}{7} \times 0.001225 \approx \frac{0.02695}{7} \approx 0.00385 \, \text{m}^2 \]

Now calculating the volume flow rate \( Q \):
\[ Q = A \cdot v = 0.00385 \times 4 \approx 0.0154 \, \text{m}^3/s \]

### Step 3: Calculate the time to fill the tank.
To find the time \( t \) to fill the tank, we use the formula:
\[ t = \frac{V}{Q} \]
Substituting the values:
\[ t = \frac{25.872}{0.0154} \approx 1685.45 \, \text{s} \]

### Step 4: Convert seconds to minutes.
To convert seconds to minutes:
\[ t_{minutes} = \frac{1685.45}{60} \approx 28.09 \, \text{minutes} \]

Thus, the time it takes to fill the tank is approximately **28 minutes**.