A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 1,907,000 gal; flow rate (Q) = 1,454,000 gal/day, determine the detention time (theta) in day.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• Everything except theta is given, so isolate theta 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: tank volume (V) = 1,907,000 gal, flow rate (Q) = 1,454,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 1.3116\ \text{day}$$

6. Step 6 — Check: returning theta = 1.3116 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 2.6231 — kept a factor of two that cancels in the correct rearrangement.
• 0.6558 — dropped that same factor in the other direction.
• 1.4427 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given flow rate (Q) = 4,483,000 gal/day; detention time (theta) = 5.4100 day, determine the tank volume (V) in gal.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: flow rate (Q) = 4,483,000 gal/day, detention time (theta) = 5.4100 day$$

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

5. Step 5 — Evaluate:

   $$V = 24253030\ \text{gal}$$

6. Step 6 — Check: returning V = 24,253,030 gal to

   $$\theta = V / Q$$

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

Why the other options are there

• 48,506,060 — kept a factor of two that cancels in the correct rearrangement.
• 12,126,515 — dropped that same factor in the other direction.
• 26,678,333 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 1,525,000 gal; detention time (theta) = 1.7500 day, determine the flow rate (Q) in gal/day.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• Everything except Q is given, so isolate Q 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: tank volume (V) = 1,525,000 gal, detention time (theta) = 1.7500 day$$

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

5. Step 5 — Evaluate:

   $$Q = 871429\ \text{gal/day}$$

6. Step 6 — Check: returning Q = 871,429 gal/day to

   $$\theta = V / Q$$

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

Why the other options are there

• 1,742,857 — kept a factor of two that cancels in the correct rearrangement.
• 435,714 — dropped that same factor in the other direction.
• 958,571 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 1,592,000 gal; flow rate (Q) = 365,000 gal/day, determine the detention time (theta) in day.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• Everything except theta is given, so isolate theta 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: tank volume (V) = 1,592,000 gal, flow rate (Q) = 365,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 4.3616\ \text{day}$$

6. Step 6 — Check: returning theta = 4.3616 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 8.7233 — kept a factor of two that cancels in the correct rearrangement.
• 2.1808 — dropped that same factor in the other direction.
• 4.7978 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given flow rate (Q) = 4,061,000 gal/day; detention time (theta) = 9.5700 day, determine the tank volume (V) in gal.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: flow rate (Q) = 4,061,000 gal/day, detention time (theta) = 9.5700 day$$

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

5. Step 5 — Evaluate:

   $$V = 38863770\ \text{gal}$$

6. Step 6 — Check: returning V = 38,863,770 gal to

   $$\theta = V / Q$$

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

Why the other options are there

• 77,727,540 — kept a factor of two that cancels in the correct rearrangement.
• 19,431,885 — dropped that same factor in the other direction.
• 42,750,147 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 1,313,000 gal; detention time (theta) = 4.4600 day, determine the flow rate (Q) in gal/day.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• Everything except Q is given, so isolate Q 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: tank volume (V) = 1,313,000 gal, detention time (theta) = 4.4600 day$$

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

5. Step 5 — Evaluate:

   $$Q = 294395\ \text{gal/day}$$

6. Step 6 — Check: returning Q = 294,395 gal/day to

   $$\theta = V / Q$$

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

Why the other options are there

• 588,789 — kept a factor of two that cancels in the correct rearrangement.
• 147,197 — dropped that same factor in the other direction.
• 323,834 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 909,000 gal; flow rate (Q) = 3,272,000 gal/day, determine the detention time (theta) in day.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• Everything except theta is given, so isolate theta 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: tank volume (V) = 909,000 gal, flow rate (Q) = 3,272,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 0.2778\ \text{day}$$

6. Step 6 — Check: returning theta = 0.2778 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 0.5556 — kept a factor of two that cancels in the correct rearrangement.
• 0.1389 — dropped that same factor in the other direction.
• 0.3056 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given flow rate (Q) = 4,673,000 gal/day; detention time (theta) = 5.1700 day, determine the tank volume (V) in gal.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: flow rate (Q) = 4,673,000 gal/day, detention time (theta) = 5.1700 day$$

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

5. Step 5 — Evaluate:

   $$V = 24159410\ \text{gal}$$

6. Step 6 — Check: returning V = 24,159,410 gal to

   $$\theta = V / Q$$

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

Why the other options are there

• 48,318,820 — kept a factor of two that cancels in the correct rearrangement.
• 12,079,705 — dropped that same factor in the other direction.
• 26,575,351 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 227,000 gal; detention time (theta) = 3.1400 day, determine the flow rate (Q) in gal/day.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• Everything except Q is given, so isolate Q 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: tank volume (V) = 227,000 gal, detention time (theta) = 3.1400 day$$

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

5. Step 5 — Evaluate:

   $$Q = 72293\ \text{gal/day}$$

6. Step 6 — Check: returning Q = 72,293 gal/day to

   $$\theta = V / Q$$

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

Why the other options are there

• 144,586 — kept a factor of two that cancels in the correct rearrangement.
• 36,146 — dropped that same factor in the other direction.
• 79,522 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 825,000 gal; flow rate (Q) = 977,000 gal/day, determine the detention time (theta) in day.

Start with the thinking

• The governing relation printed in this handbook section is Hydraulic detention time.
• Everything except theta is given, so isolate theta 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\theta = V / Q$$

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

3. Step 3

   $$List the givens: tank volume (V) = 825,000 gal, flow rate (Q) = 977,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 0.8444\ \text{day}$$

6. Step 6 — Check: returning theta = 0.8444 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 1.6888 — kept a factor of two that cancels in the correct rearrangement.
• 0.4222 — dropped that same factor in the other direction.
• 0.9289 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Settling basins following air-activated sludge reactors