water is flowing through a cylindrical pipe, of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. determine the rise in level of water in the tank in half an hour.
To determine the rise in the level of water in the tank, we need to find the volume of water that flows into the tank in half an hour.
First, let’s calculate the cross-sectional area of the pipe:
The internal diameter of the pipe is 2 cm, so the radius is 1 cm (0.01 m).
The cross-sectional area of the pipe is calculated using the formula for the area of a circle:
A = π * r^2
A = 3.14 * (0.01)^2
A ≈ 0.000314 m^2
Next, we need to calculate the volume of water that flows through the pipe per second:
The water is flowing at a rate of 0.4 m/s.
So, the volume of water flowing through the pipe per second is:
Volume = Area * Velocity
Volume = 0.000314 m^2 * 0.4 m/s
Volume ≈ 0.0001256 m^3/s
Now, let’s calculate the volume of water that flows into the tank in half an hour:
Since there are 60 minutes in an hour, half an hour is 30 minutes.
The volume of water that flows into the tank in half an hour is:
Volume = Flow rate * Time
Volume = 0.0001256 m^3/s * 30 minutes * (1/60) (conversion from minutes to hours)
Volume ≈ 0.000628 m^3
Finally, to determine the rise in the level of water in the tank, we divide the volume by the area of the base of the tank:
The radius of the tank is 40 cm, so the area of the base of the tank is:
A = π * r^2
A = 3.14 * (0.4)^2
A ≈ 0.5024 m^2
Rise in level = Volume / Area
Rise in level ≈ 0.000628 m^3 / 0.5024 m^2
Rise in level ≈ 0.00125 m
Therefore, the rise in the level of water in the tank in half an hour is approximately 0.00125 meters.