the merging of two streams to form a river. One stream has a width of 8.9 m, depth of 3.5 m, and current speed of 2.1 m/s. The other stream is 6.5 m wide and 3.7 m deep, and flows at 2.2 m/s. The width of the river is 10.7 m, and the current speed is 3.1 m/s. What is its depth?
Add the current flows in the two rivers (in cubic meters per second)
Call it Q.
Then apply Q = (width)(depth)(velocity)
and solve for the depth of the combined river.
To find the depth of the merged river, we can use the principle of continuity, which states that the product of the area and velocity of a fluid flowing through a pipe is constant.
Let's denote the depth of the merged river as D.
For the first stream:
Width = 8.9 m
Depth = 3.5 m
Current speed = 2.1 m/s
The area of the first stream can be calculated as:
Area1 = Width1 * Depth1 = 8.9 m * 3.5 m = 31.15 m^2
For the second stream:
Width = 6.5 m
Depth = 3.7 m
Current speed = 2.2 m/s
The area of the second stream can be calculated as:
Area2 = Width2 * Depth2 = 6.5 m * 3.7 m = 24.05 m^2
For the merged river:
Width = 10.7 m
Current speed = 3.1 m/s
The area of the merged river can be calculated as:
AreaTotal = WidthTotal * D = 10.7 m * D = 10.7D m^2
According to the principle of continuity, the product of the area and velocity is constant for both streams and the merged river:
Area1 * Current speed1 = AreaTotal * Current speedTotal --(1)
Area2 * Current speed2 = AreaTotal * Current speedTotal --(2)
Substituting the values into equations (1) and (2), we get:
Area1 * Current speed1 = AreaTotal * Current speedTotal
31.15 m^2 * 2.1 m/s = 10.7D m^2 * 3.1 m/s
65.415 m^3/s = 33.17D m^3/s
Area2 * Current speed2 = AreaTotal * Current speedTotal
24.05 m^2 * 2.2 m/s = 10.7D m^2 * 3.1 m/s
52.9 m^3/s = 33.17D m^3/s
Simplifying the equations, we get:
65.415 m^3/s = 33.17D m^3/s
52.9 m^3/s = 33.17D m^3/s
Since these equations are identical, we can equate them:
65.415 m^3/s = 52.9 m^3/s
Solving for D, the depth of the merged river:
33.17D = 52.9
D = 52.9 / 33.17
D ≈ 1.59 m
Therefore, the depth of the merged river is approximately 1.59 meters.
To find the depth of the merged river, we can use the principle of conservation of mass. The total mass of water entering the merged stream should be equal to the total mass of water leaving it.
Let's assume the depth of the merged river is "x" meters. We can set up the following equation based on the conservation of mass:
(Width of Stream A * Depth of Stream A * Speed of Stream A) + (Width of Stream B * Depth of Stream B * Speed of Stream B) = (Width of River * Depth of River * Speed of River)
Plugging in the given values:
(8.9 m * 3.5 m * 2.1 m/s) + (6.5 m * 3.7 m * 2.2 m/s) = 10.7 m * x * 3.1 m/s
Simplifying the equation:
65.205 m^3/s + 52.46 m^3/s = 33.17 m/s * x
117.665 m^3/s = 33.17 m/s * x
To find the value of "x", we divide both sides of the equation by 33.17 m/s:
x = 117.665 m^3/s / 33.17 m/s
Simplifying:
x ≈ 3.544 m
Therefore, the depth of the merged river would be approximately 3.544 meters.