A river 935 m wide and 3.5 m deep, flowing at 1.5 m sec-1 enters narrow canyon 55 m wide and 15 m deep. What is the average river speed in the canyon assuming its cross section is a elliptical? The area of an ellipse is π a b, where a and b are the two axes of the ellipse.

Question

A river 935 m wide and 3.5 m deep, flowing at 1.5 m sec-1 enters narrow canyon 55 m wide and 15 m deep.  What is the average river speed in the canyon assuming its cross section is a elliptical?  The area of an ellipse is π a b, where a and b are the two axes of the ellipse.

BlackTom AI · Accepted Answer

We start by recognizing the principle of conservation of flow (volume rate of flow) between the upstream river and the canyon:
Q = v1 * A1 = v2 * A2
Where:
v1 = 1.5 m/s is the upstream velocity,
A1 is the upstream cross-sectional area, and
v2 is the average speed in the canyon with cross-sectional area A2.

1) Upstream cross-sectional area A1:
The upstream cross section is essentially rectangular with width 935 m and depth 3.5 m, so A1 = 935 * 3.5 = 3272.5 m^2.

2) Canyon cross-sectional area A2:
The canyon cross section is given as elliptical with width 55 m and depth 15 m. For an ellipse, the area is A = π a b, where a and b are the semi-axes. If the width is 55 m, then the horizontal semi-axis a = 55/2 = 27.5 m. If the depth is 15 m, then the vertical semi-axis b = 15/2 = 7.5 m. Thus A2 = π * a * b = π * 27.5 * 7.5.
Compute A2: 27.5 * 7.5 = 206.25, and multiplying by π (~3.1416) gives A2 ≈ 647 m^2.

3) Compute Q using upstream values:
Q = v1 * A1 = 1.5 m/s * 3272.5 m^2 = 4908.75 m^3/s.

4) Find v2 in the canyon:
v2 = Q / A2 = 4908.75 m^3/s / 647 m^2 ≈ 7.58 m/s.

Summary:
- Upstream cross-sectional area A1 = 3272.5 m^2
- Canyon cross-sectional area A2 ≈ 647 m^2
- Flow rate Q ≈ 4908.75 m^3/s
- Canyon speed v2 ≈ 7.6 m/s

Note on the provided answer: The calculation above yields approximately 7.6 m/s. The given answer of 6 m/s appears inconsistent with the standard conservation-of-flow approach using the stated geometry (elliptical canyon cross-section with a = 27.5 m and b = 7.5 m). Depending on rounding choices or a different interpretation of the elliptical dimensions, results may vary slightly, but 7.6 m/s is the direct consequence of the stated dimensions and the continuity equation.

A river 935 m wide and 3.5 m deep, flowing at 1.5 m sec-1 enters narrow canyon 55 m wide and 15 m deep. What is the average river speed in the canyon assuming its cross section is a elliptical? The area of an ellipse is π a b, where a and b are the two axes of the ellipse.

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