A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 395.0 mg/L; rate constant (k) = 0.1400 1/day; time (t) = 7.0000 day, determine the remaining BOD (Lt) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except Lt is given, so isolate Lt 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:

   $$L_t = L_0 e^{-k t}$$

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

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 395.0 mg/L, rate constant (k) = 0.1400 1/day, time (t) = 7.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 148.2\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 148.2 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 296.5 — kept a factor of two that cancels in the correct rearrangement.
• 74.1239 — dropped that same factor in the other direction.
• 163.1 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2700 1/day; time (t) = 1.5000 day; remaining BOD (Lt) = 210.3 mg/L, determine the ultimate BOD (L0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except L0 is given, so isolate L0 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:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: rate constant (k) = 0.2700 1/day, time (t) = 1.5000 day, remaining BOD (Lt) = 210.3 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 315.3\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 315.3 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

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

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 110.0 mg/L; time (t) = 6.0000 day; remaining BOD (Lt) = 249.2 mg/L, determine the rate constant (k) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except k is given, so isolate k 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:

   $$L_t = L_0 e^{-k t}$$

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

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 110.0 mg/L, time (t) = 6.0000 day, remaining BOD (Lt) = 249.2 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = -0.1363\ \text{1/day}$$

6. Step 6 — Check: returning k = -0.1363 1/day to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• -0.2726 — kept a factor of two that cancels in the correct rearrangement.
• -0.0681 — dropped that same factor in the other direction.
• -0.1499 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 370.0 mg/L; rate constant (k) = 0.2500 1/day; time (t) = 10.0000 day, determine the remaining BOD (Lt) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except Lt is given, so isolate Lt 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:

   $$L_t = L_0 e^{-k t}$$

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

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 370.0 mg/L, rate constant (k) = 0.2500 1/day, time (t) = 10.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 30.3714\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 30.3714 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 60.7429 — kept a factor of two that cancels in the correct rearrangement.
• 15.1857 — dropped that same factor in the other direction.
• 33.4086 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.3000 1/day; time (t) = 5.5000 day; remaining BOD (Lt) = 237.0 mg/L, determine the ultimate BOD (L0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except L0 is given, so isolate L0 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:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: rate constant (k) = 0.3000 1/day, time (t) = 5.5000 day, remaining BOD (Lt) = 237.0 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 1234\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 1,234 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 2,468 — kept a factor of two that cancels in the correct rearrangement.
• 617.0 — dropped that same factor in the other direction.
• 1,357 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 170.0 mg/L; time (t) = 8.0000 day; remaining BOD (Lt) = 28.7000 mg/L, determine the rate constant (k) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except k is given, so isolate k 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:

   $$L_t = L_0 e^{-k t}$$

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

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 170.0 mg/L, time (t) = 8.0000 day, remaining BOD (Lt) = 28.7000 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.2224\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.2224 1/day to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 0.4447 — kept a factor of two that cancels in the correct rearrangement.
• 0.1112 — dropped that same factor in the other direction.
• 0.2446 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 215.0 mg/L; rate constant (k) = 0.1500 1/day; time (t) = 4.5000 day, determine the remaining BOD (Lt) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except Lt is given, so isolate Lt 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:

   $$L_t = L_0 e^{-k t}$$

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

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 215.0 mg/L, rate constant (k) = 0.1500 1/day, time (t) = 4.5000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 109.5\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 109.5 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 218.9 — kept a factor of two that cancels in the correct rearrangement.
• 54.7343 — dropped that same factor in the other direction.
• 120.4 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2700 1/day; time (t) = 2.5000 day; remaining BOD (Lt) = 389.5 mg/L, determine the ultimate BOD (L0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except L0 is given, so isolate L0 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:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: rate constant (k) = 0.2700 1/day, time (t) = 2.5000 day, remaining BOD (Lt) = 389.5 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 765.0\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 765.0 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 1,530 — kept a factor of two that cancels in the correct rearrangement.
• 382.5 — dropped that same factor in the other direction.
• 841.5 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 315.0 mg/L; time (t) = 2.0000 day; remaining BOD (Lt) = 127.4 mg/L, determine the rate constant (k) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except k is given, so isolate k 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:

   $$L_t = L_0 e^{-k t}$$

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

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 315.0 mg/L, time (t) = 2.0000 day, remaining BOD (Lt) = 127.4 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.4526\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.4526 1/day to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 0.9052 — kept a factor of two that cancels in the correct rearrangement.
• 0.2263 — dropped that same factor in the other direction.
• 0.4979 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 140.0 mg/L; rate constant (k) = 0.2800 1/day; time (t) = 7.0000 day, determine the remaining BOD (Lt) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except Lt is given, so isolate Lt 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:

   $$L_t = L_0 e^{-k t}$$

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

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 140.0 mg/L, rate constant (k) = 0.2800 1/day, time (t) = 7.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 19.7202\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 19.7202 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 39.4404 — kept a factor of two that cancels in the correct rearrangement.
• 9.8601 — dropped that same factor in the other direction.
• 21.6922 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Cyclone Effective Number of Turns Approximation