water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. in what time will the level of water in pond rise by 21 cm? what should be the speed of water if the rise in water level is to be attained in 1 hour?
Water Flow Rate and Time to Rise Water Level Calculations:
Given Data:
- Water flow rate: 15 km/h
- Pipe diameter: 14 cm
- Cuboidal pond dimensions: 50 m long, 44 m wide
- Desired rise in water level: 21 cm
Calculating Volume of Water Flowing:
First, calculate the volume of water flowing through the pipe per hour:
- Convert the water flow rate to m/s: 15 \, km/h = \frac{15 \times 1000}{3600} = \frac{15000}{3600} \approx 4.17 \, m/s
- Calculate the cross-sectional area of the pipe: r = \frac{d}{2} = \frac{14}{2} = 7 \, cm = 0.07 \, m
- A = \pi r^2 = \pi \times (0.07)^2 \approx 0.0154 \, m^2
- Volume of water flowing per second: V = 0.0154 \times 4.17 \approx 0.0643 \, m^3/s
- Volume of water flowing per hour: V_{hour} = 0.0643 \times 3600 = 231.48 \, m^3/h
Calculating Time to Rise Water Level by 21 cm:
- The volume of water required to rise by 21 cm in the cuboidal pond: V_{rise} = 50 \times 44 \times 0.21 = 462 \, m^3
- Time required to achieve this rise: \frac{V_{rise}}{V_{hour}} = \frac{462}{231.48} = 1.996 \, hours
Therefore, it will take approximately 1.996 hours for the water level in the pond to rise by 21 cm.
Calculating Required Speed for 1 Hour Rise:
To achieve the desired rise in water level within 1 hour:
- Volume of water required in 1 hour: V_{1hour} = V_{hour} = 231.48 \, m^3
- Volume of water required to rise by 21 cm in 1 hour: V_{rise\_1hour} = 50 \times 44 \times 0.21 = 462 \, m^3
- Speed of water required: \frac{V_{rise\_1hour}}{3600} = \frac{462}{3600} \approx 0.1283 \, m^3/s
Therefore, the speed of water needs to be approximately 0.1283 m/s to attain a rise in the water level by 21 cm in 1 hour.