---
title: "Most economical trapezoidal channel of area 50 m²"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:economical-trapezoidal-area-50"
source: https://www.bctengineer.com/q/semester_6/hydro_power/economical-trapezoidal-area-50
site: BCT Engineering
---

# Most economical trapezoidal channel of area 50 m²

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2080 Chaitra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/economical-trapezoidal-area-50
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

A trapezoidal channel has side slopes of 1:2 (H:V) and slope of the bed is 1 in 1500. The sectional area is 50 m². Find the dimensions of the section if it is considered the most economical. Also determine the discharge if Chezy's C = 50.

## Short answer

For the most economical trapezoid, half top width equals one sloping side and R = y/2. With A=50 m², side slope 1:2 (H:V), bed slope 1/1500, solution gives bottom width 6.63 m, depth 5.37 m. Chezy's C=50 yields discharge 105.7 m³/s.

## Answer

For the most economical trapezoidal section: half the top width equals one sloping side, and $R = y/2$.

#### Data
Side slope 1 : 2 (H : V), so horizontal per unit vertical $n = 0.5$; $S = 1/1500$; $A = 50$ m²; $C = 50$.

#### Condition and dimensions
$$
\begin{aligned}
\frac{b + 2ny}{2} &= y\sqrt{1+n^2} \\
b &= 2y\sqrt{1+0.25} - 2(0.5)y = (2.2361 - 1)y = 1.2361y \\
A &= (b + ny)y = (1.2361 + 0.5)y^2 = 1.7361y^2 \\
50 &= 1.7361y^2 \;\Rightarrow\; y = 5.367\ \text{m} \\
b &= 1.2361 \times 5.367 = 6.634\ \text{m}
\end{aligned}
$$

Check: $A = (6.634 + 0.5 \times 5.367) \times 5.367 = 9.317 \times 5.367 = 50.0$ m². Wetted perimeter $P = 6.634 + 2 \times 5.367 \times 1.118 = 18.63$ m, $R = 50/18.63 = 2.683$ m $= y/2$.

#### Discharge (Chezy)
$$
\begin{aligned}
Q &= AC\sqrt{RS} = 50 \times 50 \times \sqrt{2.683 \times \frac{1}{1500}} \\
&= 2500 \times 0.04229 = 105.7\ \text{m}^3/\text{s}
\end{aligned}
$$

```text
  |<------ T = 12.00 m ------->|
  \                            /
   \     y = 5.37 m     1H:2V /
    \________________________/
          b = 6.63 m
```

**Answer:** Bottom width ≈ 6.63 m, depth ≈ 5.37 m, discharge ≈ 105.7 m³/s (velocity ≈ 2.11 m/s).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A side slope of 1:2 (H:V) means n equals:
   A. 2
   B. 0.5
   C. 1
   D. 0.25
   Answer: B. 1 horizontal per 2 vertical gives n = 0.5 horizontal per unit vertical. (Key point: Side slope meaning)

2. What gives b in terms of y for the most economical section?
   A. b = 2y(sqrt(1+n^2) - n) = 1.2361 y
   B. b = 2y
   C. b = y sqrt(1+n^2)
   D. b = y/2
   Answer: A. From (b+2ny)/2 = y sqrt(1+n^2): b = 2y(1.1180 - 0.5) = 1.2361 y. (Key point: Economical condition)

3. With A = 1.7361 y^2 = 50, the depth is:
   A. y = 3.50 m
   B. y = 5.37 m
   C. y = 7.00 m
   D. y = 28.8 m
   Answer: B. y = sqrt(50/1.7361) = 5.367 m and b = 6.63 m. (Key point: Depth from area)

4. Which quantity is used for R in the Chezy discharge?
   A. R = y = 5.37 m
   B. R = y/2 = 2.68 m
   C. R = A/b
   D. R = P/A
   Answer: B. R = y/2 = 2.683 m, so Q = 50 x 50 x sqrt(2.683/1500) = 105.7 m^3/s. (Key point: Discharge)

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