Water is flowing at the rate of 5 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 25 m long and 22 m wide. Determine the time in which the level of water in the tank will rise by 21 cm. (take`pi` = 22/7)

Answer

To solve the problem, we need to determine the time it takes for the water level in the tank to rise by 21 cm when water flows through a pipe. Here’s the step-by-step solution:

### Step 1: Convert the flow rate into meters per second
The flow rate of water is given as 5 km/h. We need to convert this to meters per second (m/s).

\[ \text{Flow rate in m/s} = \frac{5 \times 1000 \text{ m}}{3600 \text{ s}} = \frac{5000}{3600} \approx 1.39 \text{ m/s} \]

**Hint:** To convert km/h to m/s, multiply by \( \frac{1000}{3600} \).

### Step 2: Calculate the radius of the pipe
The diameter of the pipe is given as 14 cm. We need to find the radius in meters.

\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} = 0.07 \text{ m} \]

**Hint:** Remember that the radius is half of the diameter, and convert cm to m by dividing by 100.

### Step 3: Calculate the cross-sectional area of the pipe
The cross-sectional area \( A \) of the pipe can be calculated using the formula for the area of a circle, \( A = \pi r^2 \).

\[ A = \frac{22}{7} \times (0.07)^2 = \frac{22}{7} \times 0.0049 \approx 0.154 \text{ m}^2 \]

**Hint:** Use \( \pi \approx \frac{22}{7} \) for calculations involving circles.

### Step 4: Calculate the volume of water flowing through the pipe per second
Now, we can find the volume of water flowing through the pipe per second using the formula \( \text{Volume} = \text{Area} \times \text{Flow rate} \).

\[ \text{Volume per second} = A \times \text{Flow rate} = 0.154 \text{ m}^2 \times 1.39 \text{ m/s} \approx 0.214 \text{ m}^3/\text{s} \]

**Hint:** Volume can be found by multiplying the cross-sectional area by the flow speed.

### Step 5: Calculate the volume of water needed to raise the tank's water level by 21 cm
The tank is rectangular, and we can find the volume needed to raise the water level by 21 cm (0.21 m).

\[ \text{Volume of water needed} = \text{Length} \times \text{Width} \times \text{Height} = 25 \text{ m} \times 22 \text{ m} \times 0.21 \text{ m} = 110.25 \text{ m}^3 \]

**Hint:** The volume of a rectangular prism is calculated by multiplying its length, width, and height.

### Step 6: Calculate the time taken to fill this volume
Finally, we can find the time taken to fill the tank by dividing the total volume needed by the volume flow rate.

\[ \text{Time} = \frac{\text{Volume of water needed}}{\text{Volume per second}} = \frac{110.25 \text{ m}^3}{0.214 \text{ m}^3/\text{s}} \approx 515.9 \text{ seconds} \]

**Hint:** Time can be calculated by dividing the total volume by the rate of flow.

### Final Answer
The time taken for the water level in the tank to rise by 21 cm is approximately **516 seconds** or about **8.6 minutes**.

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