A 3.0 m wide rectangular channel flows 1.2 m deep on a 0.0016 slope with n = 0.015. Find the discharge.

Start with the thinking

• Hydraulic radius uses the wetted perimeter — two walls plus the bed.
• Metric Manning uses the 1.00 coefficient, not 1.49.

Step-by-step solution

1. Area

   $$A = by = 3.0(1.2) = 3.60 m^{2}$$

2. Wetted perimeter

   $$P = b + 2y = 3.0 + 2.4 = 5.40 m$$

3. Hydraulic radius

   $$R = A/P = 3.60/5.40 = 0.667 m$$

4. Manning

   $$Q = (1/n)AR^{2/3}S^{1/2}$$

5. Terms

   $$R^{2/3} = 0.763 and S^{1/2} = 0.0400$$

6. Substitute

   $$Q = (1/0.015)(3.60)(0.763)(0.0400)$$

7. Result

   $$Q = 7.32 m^{3}/s$$

Why the other options are there

• 10.9 m³/s (1.49 coefficient used in SI)
• 5.49 m³/s (R taken as the depth)

Reference: FE Reference Handbook — Hydraulics — Manning's equation

A hydraulics problem uses Manning's equation. Given Manning's roughness (n) = 0.0500; hydraulic radius (R) = 5.5000 ft; channel slope (S) = 0.0054 ft/ft, determine the velocity (V) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except V is given, so isolate V symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3 — List the givens: Manning's roughness (n) = 0.0500, hydraulic radius (R) = 5.5000 ft, channel slope (S) = 0.0054 ft/ft.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 6.8049\ \text{ft/s}$$

6. Step 6 — Check: returning V = 6.8049 ft/s to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 13.6098 — kept a factor of two that cancels in the correct rearrangement.
• 3.4024 — dropped that same factor in the other direction.
• 7.4854 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given hydraulic radius (R) = 4.6000 ft; channel slope (S) = 0.0134 ft/ft; velocity (V) = 13.4300 ft/s, determine the Manning's roughness (n).

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3 — List the givens: hydraulic radius (R) = 4.6000 ft, channel slope (S) = 0.0134 ft/ft, velocity (V) = 13.4300 ft/s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 0.0354$$

6. Step 6 — Check: returning n = 0.0354 to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0709 — kept a factor of two that cancels in the correct rearrangement.
• 0.0177 — dropped that same factor in the other direction.
• 0.0390 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given Manning's roughness (n) = 0.0150; hydraulic radius (R) = 3.2000 ft; velocity (V) = 12.9100 ft/s, determine the channel slope (S) in ft/ft.

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except S is given, so isolate S symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that S stands alone on the left-hand side.

3. Step 3 — List the givens: Manning's roughness (n) = 0.0150, hydraulic radius (R) = 3.2000 ft, velocity (V) = 12.9100 ft/s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$S = 0.0036\ \text{ft/ft}$$

6. Step 6 — Check: returning S = 0.0036 ft/ft to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0072 — kept a factor of two that cancels in the correct rearrangement.
• 0.0018 — dropped that same factor in the other direction.
• 0.0040 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given Manning's roughness (n) = 0.0340; hydraulic radius (R) = 3.6000 ft; channel slope (S) = 0.0108 ft/ft, determine the velocity (V) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except V is given, so isolate V symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3 — List the givens: Manning's roughness (n) = 0.0340, hydraulic radius (R) = 3.6000 ft, channel slope (S) = 0.0108 ft/ft.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 10.6689\ \text{ft/s}$$

6. Step 6 — Check: returning V = 10.6689 ft/s to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 21.3378 — kept a factor of two that cancels in the correct rearrangement.
• 5.3345 — dropped that same factor in the other direction.
• 11.7358 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given hydraulic radius (R) = 5.5000 ft; channel slope (S) = 0.0138 ft/ft; velocity (V) = 9.9100 ft/s, determine the Manning's roughness (n).

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3 — List the givens: hydraulic radius (R) = 5.5000 ft, channel slope (S) = 0.0138 ft/ft, velocity (V) = 9.9100 ft/s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 0.0549$$

6. Step 6 — Check: returning n = 0.0549 to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.1098 — kept a factor of two that cancels in the correct rearrangement.
• 0.0274 — dropped that same factor in the other direction.
• 0.0604 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given Manning's roughness (n) = 0.0480; hydraulic radius (R) = 1.9000 ft; velocity (V) = 4.2500 ft/s, determine the channel slope (S) in ft/ft.

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except S is given, so isolate S symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that S stands alone on the left-hand side.

3. Step 3 — List the givens: Manning's roughness (n) = 0.0480, hydraulic radius (R) = 1.9000 ft, velocity (V) = 4.2500 ft/s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$S = 0.0080\ \text{ft/ft}$$

6. Step 6 — Check: returning S = 0.0080 ft/ft to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0160 — kept a factor of two that cancels in the correct rearrangement.
• 0.0040 — dropped that same factor in the other direction.
• 0.0088 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given Manning's roughness (n) = 0.0440; hydraulic radius (R) = 1.0000 ft; channel slope (S) = 0.0029 ft/ft, determine the velocity (V) in ft/s.

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except V is given, so isolate V symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3 — List the givens: Manning's roughness (n) = 0.0440, hydraulic radius (R) = 1.0000 ft, channel slope (S) = 0.0029 ft/ft.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 1.8187\ \text{ft/s}$$

6. Step 6 — Check: returning V = 1.8187 ft/s to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.6374 — kept a factor of two that cancels in the correct rearrangement.
• 0.9094 — dropped that same factor in the other direction.
• 2.0006 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given hydraulic radius (R) = 4.3000 ft; channel slope (S) = 0.0059 ft/ft; velocity (V) = 14.2600 ft/s, determine the Manning's roughness (n).

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except n is given, so isolate n symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that n stands alone on the left-hand side.

3. Step 3 — List the givens: hydraulic radius (R) = 4.3000 ft, channel slope (S) = 0.0059 ft/ft, velocity (V) = 14.2600 ft/s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$n = 0.0212$$

6. Step 6 — Check: returning n = 0.0212 to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0423 — kept a factor of two that cancels in the correct rearrangement.
• 0.0106 — dropped that same factor in the other direction.
• 0.0233 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation

A hydraulics problem uses Manning's equation. Given Manning's roughness (n) = 0.0420; hydraulic radius (R) = 3.4000 ft; velocity (V) = 5.1700 ft/s, determine the channel slope (S) in ft/ft.

Start with the thinking

• The governing relation printed in this handbook section is Manning's equation.
• Everything except S is given, so isolate S symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Hydraulics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

2. Step 2 — Rearrange the relation so that S stands alone on the left-hand side.

3. Step 3 — List the givens: Manning's roughness (n) = 0.0420, hydraulic radius (R) = 3.4000 ft, velocity (V) = 5.1700 ft/s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$S = 0.0042\ \text{ft/ft}$$

6. Step 6 — Check: returning S = 0.0042 ft/ft to

   $$V = (1.486/n) R^{2/3} S^{1/2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0084 — kept a factor of two that cancels in the correct rearrangement.
• 0.0021 — dropped that same factor in the other direction.
• 0.0046 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Hydraulics → Manning's Equation