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 Calculation and Rising Time of Water Level:
1. Calculate the Time for Water Level to Rise by 21 cm:
The volume of water flowing through the pipe per second can be calculated using the formula:
\text{Volume} = \pi \times (\text{radius})^2 \times \text{speed}
Given that the diameter of the pipe is 14 cm, the radius ((r)) is half of the diameter, so (r = 7) cm.
Converting the speed from km/h to cm/s gives 15 \, \text{km/h} = 1500000 \, \text{cm/h} = 416.67 \, \text{cm/s}.
Substitute these values into the formula:
\text{Volume} = \pi \times 7^2 \times 416.67 = 78540.56\pi \, \text{cm}^3/s
Now, to raise the water level by 21 cm in the cuboidal pond, we can calculate the time needed using the formula:
\text{Time} = \frac{\text{Volume of Water}}{\text{Area of the Pond Base} \times \text{Rise in Water Level}}
Given that the pond is 50 m long and 44 m wide, the area of the base is 50 \times 44 \times 10000 = 2200000 \, \text{cm}^2.
Substitute the values to find the time needed:
\text{Time} = \frac{78540.56\pi}{2200000 \times 21} \approx 5.98 \, \text{seconds}
Therefore, the water level in the pond will rise by 21 cm in approximately 5.98 seconds.
2. Calculate the Required Speed of Water for the Rise in 1 Hour:
If the rise in water level is to be attained in 1 hour, which is 3600 seconds, we need to find the speed required.
Using the same formula as above to calculate the volume of water flowing per second, we have:
\text{Volume} = 78540.56\pi \, \text{cm}^3/s
To find the speed needed:
\text{Speed} = \frac{\text{Volume}}{\pi \times 7^2} = \frac{78540.56\pi}{49\pi} \approx 1601.86 \, \text{cm/s}
Therefore, the speed of water needs to be approximately 1601.86 cm/s for the water level to rise by 21 cm in 1 hour.