---
title: "Most economical trapezoid of area 40 m²"
subject: "Hydraulics"
chapter: 6
key: "semester_4/hydraulics:economical-trapezoidal-section-area-40"
source: https://www.bctengineer.com/q/semester_4/hydraulics/economical-trapezoidal-section-area-40
site: BCT Engineering
---

# Most economical trapezoid of area 40 m²

Hydraulics · Chapter 6: Uniform flow in open channel

- Asked in: 2075 Bhadra, 5 marks
- Video: [Most Economical Trapezoidal Channel Numericals || Open Channel Flow Numericals || Hydraulics || OCF](https://www.youtube.com/watch?v=71qj_xrEwmk) — Mukesh Kumar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/economical-trapezoidal-section-area-40
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-6

## Question

A trapezoidal channel has side slope 1:2 (H:V) and the slope of the bed is 1 in 1500. The area of the section is 40 m². Find the dimensions if it is most economical. Determine the discharge of the most economical section if Chezy's constant (C) = 50.

## Short answer

With z=0.5 and area 40 m², most economical depth y = 4.80 m, bed width b = 5.93 m. Using C=50 and bed slope 1/1500, discharge Q = 80.0 m³/s.

## Answer

**Reading of the side slope:** "1:2 (H:V)" is taken as 1 horizontal to 2 vertical, so $z=0.5$.

#### Given data
$A=40$ m², $z=0.5$, $S_0=1/1500$, $C=50$.

#### Most economical trapezoidal section
Conditions: $R=\dfrac y2$ and $b=2y(\sqrt{1+z^2}-z)$.
$$
A=(b+zy)y=y^2\left(2\sqrt{1+z^2}-z\right)=y^2\left(2\sqrt{1.25}-0.5\right)=1.7361\,y^2
$$
$$
y=\sqrt{\frac{40}{1.7361}}=4.800\ \text{m}
$$
$$
b=2y(\sqrt{1.25}-0.5)=2\times4.800\times0.6180=5.933\ \text{m}
$$
Check: $A=(5.933+0.5\times4.800)\times4.800=40.0$ m².

Wetted perimeter $P=b+2y\sqrt{1+z^2}=16.666$ m; $R=A/P=2.400$ m $=y/2$ (✓).

#### Discharge
$$
V=C\sqrt{RS_0}=50\sqrt{\frac{2.400}{1500}}=2.000\ \text{m/s}
$$
$$
Q=AV=40\times2.000=80.00\ \text{m}^3/\text{s}
$$
**Answer: depth $y=4.80$ m, bed width $b=5.93$ m, discharge $Q=80.0$ m³/s.**

## Check yourself

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

1. Which pair of conditions defines the most economical trapezoidal section?
   A. R = y and b = y
   B. R = y/2 and b = 2y(sqrt(1+z^2) - z)
   C. R = 2y and b = y/z
   D. R = y/2 and b = 2y
   Answer: B. The best trapezoid has R = y/2, and its bed width follows b = 2y(sqrt(1+z^2) - z). (Key point: Conditions for economical trapezoid)

2. The side slope is 1:2 (H:V). What value of z (horizontal per vertical) is used?
   A. z = 2
   B. z = 1
   C. z = 0.5
   D. z = 1.5
   Answer: C. 1 horizontal to 2 vertical means z = 0.5. (Key point: Reading of the side slope)

3. How is the depth obtained for A = 40 m2?
   A. From A = y^2(2 sqrt(1+z^2) - z) = 1.7361 y^2, so y = 4.80 m
   B. From A = 2y^2, so y = 4.47 m
   C. From A = y^2 (z + 1), so y = 5.16 m
   D. From A = 40 = b + y, so y = 20 m
   Answer: A. For the economical trapezoid with z = 0.5, A = 1.7361 y^2, giving y = sqrt(40/1.7361) = 4.80 m. (Key point: Finding the depth)

4. With C = 50, S0 = 1/1500 and R = y/2 = 2.4 m, the discharge is:
   A. V = 2.0 m/s and Q = 80 m3/s
   B. V = 4.0 m/s and Q = 160 m3/s
   C. V = 1.0 m/s and Q = 40 m3/s
   D. V = 2.0 m/s and Q = 16.7 m3/s
   Answer: A. V = C sqrt(R S0) = 50 sqrt(2.4/1500) = 2.0 m/s, and Q = A V = 40 x 2 = 80 m3/s. (Key point: Discharge with Chezy)

## Related in Hydraulics

- [Economical trapezoid and power saving](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aeconomical-trapezoid-saving-in-power-vs-rectangular) — Hydraulics, chapter 6
- [Economical trapezoid for given velocity](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aeconomical-trapezoidal-velocity-0-6-bed-slope) — Hydraulics, chapter 6
- [Economical channel: unlined versus concrete lined](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aeconomical-trapezoidal-unlined-vs-lined) — Hydraulics, chapter 6, asked 2×
- [Efficient trapezoid with unequal side slopes](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aefficient-trapezoidal-two-side-slopes-critical-depth) — Hydraulics, chapter 6
- [Half-hexagon channel dimensions and Manning n](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahalf-hexagon-channel-chezy-manning-n) — Hydraulics, chapter 6
- [Derive most economical rectangular section](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Amost-economical-rectangular-section-derive) — Hydraulics, chapter 6

## Related in other subjects

- [Most economical trapezoidal channel of area 50 m²](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeconomical-trapezoidal-area-50) — Hydro Power, chapter 1
- [Economical rectangular condition and trapezoidal bed slope](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atrapezoidal-channel-bed-slope) — Hydro Power, chapter 1
- [Economical trapezoidal channel side length](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeconomical-trapezoidal-side-length) — Hydro Power, chapter 1
- [Most efficient trapezoidal channel (3H:4V)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aefficient-trapezoidal-3h4v) — Hydro Power, chapter 1