A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 525,000 gal; flow rate (Q) = 2,301,000 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

   $$List the givens: tank volume (V) = 525,000 gal, flow rate (Q) = 2,301,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 0.2282\ \text{day}$$

6. Step 6 — Check: returning theta = 0.2282 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 0.4563 — kept a factor of two that cancels in the correct rearrangement.
• 0.1141 — dropped that same factor in the other direction.
• 0.2510 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given flow rate (Q) = 3,155,000 gal/day; detention time (theta) = 5.0700 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

   $$List the givens: flow rate (Q) = 3,155,000 gal/day, detention time (theta) = 5.0700 day$$

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

5. Step 5 — Evaluate:

   $$V = 15995850\ \text{gal}$$

6. Step 6 — Check: returning V = 15,995,850 gal to

   $$\theta = V / Q$$

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

Why the other options are there

• 31,991,700 — kept a factor of two that cancels in the correct rearrangement.
• 7,997,925 — dropped that same factor in the other direction.
• 17,595,435 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 474,000 gal; detention time (theta) = 8.4400 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

   $$List the givens: tank volume (V) = 474,000 gal, detention time (theta) = 8.4400 day$$

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

5. Step 5 — Evaluate:

   $$Q = 56161\ \text{gal/day}$$

6. Step 6 — Check: returning Q = 56,161 gal/day to

   $$\theta = V / Q$$

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

Why the other options are there

• 112,322 — kept a factor of two that cancels in the correct rearrangement.
• 28,081 — dropped that same factor in the other direction.
• 61,777 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 1,230,000 gal; flow rate (Q) = 3,336,000 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

   $$List the givens: tank volume (V) = 1,230,000 gal, flow rate (Q) = 3,336,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 0.3687\ \text{day}$$

6. Step 6 — Check: returning theta = 0.3687 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 0.7374 — kept a factor of two that cancels in the correct rearrangement.
• 0.1844 — dropped that same factor in the other direction.
• 0.4056 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given flow rate (Q) = 1,391,000 gal/day; detention time (theta) = 4.9600 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

   $$List the givens: flow rate (Q) = 1,391,000 gal/day, detention time (theta) = 4.9600 day$$

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

5. Step 5 — Evaluate:

   $$V = 6899360\ \text{gal}$$

6. Step 6 — Check: returning V = 6,899,360 gal to

   $$\theta = V / Q$$

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

Why the other options are there

• 13,798,720 — kept a factor of two that cancels in the correct rearrangement.
• 3,449,680 — dropped that same factor in the other direction.
• 7,589,296 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 972,000 gal; detention time (theta) = 3.7400 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

   $$List the givens: tank volume (V) = 972,000 gal, detention time (theta) = 3.7400 day$$

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

5. Step 5 — Evaluate:

   $$Q = 259893\ \text{gal/day}$$

6. Step 6 — Check: returning Q = 259,893 gal/day to

   $$\theta = V / Q$$

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

Why the other options are there

• 519,786 — kept a factor of two that cancels in the correct rearrangement.
• 129,947 — dropped that same factor in the other direction.
• 285,882 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 803,000 gal; flow rate (Q) = 3,747,000 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

   $$List the givens: tank volume (V) = 803,000 gal, flow rate (Q) = 3,747,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 0.2143\ \text{day}$$

6. Step 6 — Check: returning theta = 0.2143 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 0.4286 — kept a factor of two that cancels in the correct rearrangement.
• 0.1072 — dropped that same factor in the other direction.
• 0.2357 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given flow rate (Q) = 4,766,000 gal/day; detention time (theta) = 7.7800 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

   $$List the givens: flow rate (Q) = 4,766,000 gal/day, detention time (theta) = 7.7800 day$$

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

5. Step 5 — Evaluate:

   $$V = 37079480\ \text{gal}$$

6. Step 6 — Check: returning V = 37,079,480 gal to

   $$\theta = V / Q$$

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

Why the other options are there

• 74,158,960 — kept a factor of two that cancels in the correct rearrangement.
• 18,539,740 — dropped that same factor in the other direction.
• 40,787,428 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 1,297,000 gal; detention time (theta) = 0.2700 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

   $$List the givens: tank volume (V) = 1,297,000 gal, detention time (theta) = 0.2700 day$$

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

5. Step 5 — Evaluate:

   $$Q = 4803704\ \text{gal/day}$$

6. Step 6 — Check: returning Q = 4,803,704 gal/day to

   $$\theta = V / Q$$

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

Why the other options are there

• 9,607,407 — kept a factor of two that cancels in the correct rearrangement.
• 2,401,852 — dropped that same factor in the other direction.
• 5,284,074 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage

A environmental engineering problem uses Hydraulic detention time. Given tank volume (V) = 1,960,000 gal; flow rate (Q) = 668,000 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

   $$List the givens: tank volume (V) = 1,960,000 gal, flow rate (Q) = 668,000 gal/day$$

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

5. Step 5 — Evaluate:

   $$\theta = 2.9341\ \text{day}$$

6. Step 6 — Check: returning theta = 2.9341 day to

   $$\theta = V / Q$$

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

Why the other options are there

• 5.8683 — kept a factor of two that cancels in the correct rearrangement.
• 1.4671 — dropped that same factor in the other direction.
• 3.2275 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Second Stage