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 identifying the given geometry and flow information. The question asks for the average speed in the canyon, assuming the cross section is elliptical and that the flow rate is conserved.

Step 1: Compute the cross-sectional area of the river before the canyon. For an ellipse, A = π a b, where a and b are the semi-axes. The river width 935 m gives a = 935/2 = 467.5 m, and the depth 3.5 m gives b = 3.5/2 = 1.75 m. Thus A_river = π × 467.5 × 1.75.

Step 2: Determine the flow rate (volume per unit time) in the main river. With a flow speed of 1.5 m/s, Q = v × A = 1.5 × (π × 467.5 × 1.75).

Step 3: Compute the cross-sectional area of the canyon. The canyon has width 55 m and depth 15 m, so the ellipse has semi-axes a = 55/2 = 27.5 m and b = 15/2 = 7.5 m. Therefore A_canyon = π × 27.5 × 7.5.

Step 4: Use conservation of mass (incompressible flow) to find the speed in the canyon. The same Q must pass through the canyon, so v_canyon = Q / A_canyon = [1.5 × (π × 467.5 × 1.75)] ÷ (π × 27.5 × 7.5).

Step 5: Simplify and compute. The π terms cancel, leaving v_canyon = 1.5 × (467.5 × 1.75) / (27.5 × 7.5).

Step 6: Perform the arithmetic in steps. 467.5 × 1.75 = 817.8125. The numerator becomes 1.5 × 817.8125 = 1226.71875. The denominator 27.5 × 7.5 = 206.25.

Step 7: Divide the results: v_canyon ≈ 1226.71875 ÷ 206.25 ≈ 5.95 m/s. Rounding as appropriate gives a value around 6 m/s.

Summary: By treating both cross sections as ellipses and applying conservation of flow, the canyon speed comes out to roughly 5.95 m/s, which is approximately 6 m/s when rounded.

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.数值题

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