To find out how long it takes to fill the cylindrical water tank, we need to follow these steps:
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Calculate the volume of the tank: The formula for the volume of a cylinder is: \[ \text{Volume} = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
- The diameter of the tank is 14 m, so the radius \( r \) is half of the diameter: \[ r = \frac{14}{2} = 7 \text{ m} \]
- The height \( h \) of the tank is given as 7 m.
- Using \( \pi \) as \( \frac{22}{7} \): \[ \text{Volume of the tank} = \frac{22}{7} \times (7)^2 \times (7) \] \[ = \frac{22}{7} \times 49 \times 7 \] \[ = \frac{22 \times 49 \times 7}{7} \] \[ = 22 \times 49 \] \[ = 1078 \text{ m}^3 \]
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Calculate the flow rate from the pipe: The flow rate can be calculated using the area of the pipe and the velocity of water.
- The radius of the pipe is given as 70 mm, which we convert to meters: \[ r_{\text{pipe}} = 70 \text{ mm} = 0.07 \text{ m} \]
- The cross-sectional area \( A \) of the pipe is given by: \[ A = \pi r_{\text{pipe}}^2 \] \[ = \frac{22}{7} \times (0.07)^2 \] \[ = \frac{22}{7} \times 0.0049 \] \[ = \frac{22 \times 0.0049}{7} \] \[ = \frac{0.1078}{7} \] \[ \approx 0.0154 \text{ m}^2 \]
- The flow rate \( Q \) is given by: \[ Q = \text{Area} \times \text{velocity} \] \[ = 0.0154 \text{ m}^2 \times 7 \text{ m/s} \] \[ = 0.1078 \text{ m}^3/\text{s} \]
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Calculate the time to fill the tank: To find the time \( t \) required to fill the tank, use: \[ t = \frac{\text{Volume}}{\text{Flow Rate}} \] \[ = \frac{1078 \text{ m}^3}{0.1078 \text{ m}^3/\text{s}} \] \[ \approx 9995.16 \text{ seconds} \]
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Convert seconds to hours: To convert seconds into hours, divide by the number of seconds in an hour (3600 seconds): \[ \text{Time in hours} = \frac{9995.16 \text{ seconds}}{3600 \text{ seconds/hour}} \] \[ \approx 2.77 \text{ hours} \]
Therefore, it takes approximately 2.77 hours to fill the tank completely.
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