Q.14 A trapezoidal canal carries a discharge of 40 cumec and has the permissible mean velocity of 0.95 m/s. The bed width of the canal is _______. Take the side slope of the canal as 1:1 and the width/depth ratio as 6.5.
- A17.40 m
- B14.50 m
- C16.40 m
- D15.40 mCorrect
Solution
Concept
Area = Q/V. For a trapezoid with side slope 1:1, A = (b + d) d. Use b = 6.5 d.
Formula
A = Q/V; A = (b + z d) d; b = 6.5 d
Step-by-step solution
1. A = Q/V = 40/0.95 = 42.105 m². 2. Side slope 1:1 means z = 1, so A = (b + d) d. 3. Put b = 6.5 d: A = (6.5 d + d) d = 7.5 d2. 4. 7.5 d2 = 42.105, so d2 = 5.614 and d = 2.369 m. 5. b = 6.5 × 2.369 = 15.40 m.
Explanation
First find the flow area from discharge and velocity. Then write the trapezoid area in terms of depth only, using b = 6.5 d. Solving gives d ≈ 2.37 m, so the bed width b ≈ 15.40 m.
Why the other options are wrong
A (17.40 m), B (14.50 m) and C (16.40 m) do not satisfy A = 42.105 m² with b/d = 6.5 and side slope 1:1.
Shortcut
A = (n + z) d2 = 7.5 d2. So d = √(42.105/7.5) ≈ 2.37 m, and b = 6.5 d ≈ 15.4 m.
Common mistake
Using A = b d (rectangle) or forgetting the side slope term z d2.
Correct answer: D) 15.40 m