A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 150.0 mg/L; rate constant (k) = 0.2800 1/day; time (t) = 8.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 150.0 mg/L, rate constant (k) = 0.2800 1/day, time (t) = 8.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 15.9688\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 15.9688 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

• 31.9376 — kept a factor of two that cancels in the correct rearrangement.
• 7.9844 — dropped that same factor in the other direction.
• 17.5657 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1700 1/day; time (t) = 8.0000 day; remaining BOD (Lt) = 40.4000 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

   $$List the givens: rate constant (k) = 0.1700 1/day, time (t) = 8.0000 day, remaining BOD (Lt) = 40.4000 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 157.4\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 157.4 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

• 314.8 — kept a factor of two that cancels in the correct rearrangement.
• 78.7031 — dropped that same factor in the other direction.
• 173.1 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 215.0 mg/L; time (t) = 5.0000 day; remaining BOD (Lt) = 15.7000 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

   $$List the givens: ultimate BOD (L_{0}) = 215.0 mg/L, time (t) = 5.0000 day, remaining BOD (Lt) = 15.7000 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.5234\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.5234 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

• 1.0468 — kept a factor of two that cancels in the correct rearrangement.
• 0.2617 — dropped that same factor in the other direction.
• 0.5757 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 355.0 mg/L; rate constant (k) = 0.0600 1/day; time (t) = 8.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 355.0 mg/L, rate constant (k) = 0.0600 1/day, time (t) = 8.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 219.7\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 219.7 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

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

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1900 1/day; time (t) = 5.0000 day; remaining BOD (Lt) = 274.6 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

   $$List the givens: rate constant (k) = 0.1900 1/day, time (t) = 5.0000 day, remaining BOD (Lt) = 274.6 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 710.0\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 710.0 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

• 1,420 — kept a factor of two that cancels in the correct rearrangement.
• 355.0 — dropped that same factor in the other direction.
• 781.0 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 205.0 mg/L; time (t) = 1.5000 day; remaining BOD (Lt) = 283.9 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

   $$List the givens: ultimate BOD (L_{0}) = 205.0 mg/L, time (t) = 1.5000 day, remaining BOD (Lt) = 283.9 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = -0.2171\ \text{1/day}$$

6. Step 6 — Check: returning k = -0.2171 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

• -0.4341 — kept a factor of two that cancels in the correct rearrangement.
• -0.1085 — dropped that same factor in the other direction.
• -0.2388 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 220.0 mg/L; rate constant (k) = 0.1500 1/day; time (t) = 2.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 220.0 mg/L, rate constant (k) = 0.1500 1/day, time (t) = 2.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 163.0\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 163.0 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

• 326.0 — kept a factor of two that cancels in the correct rearrangement.
• 81.4900 — dropped that same factor in the other direction.
• 179.3 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2400 1/day; time (t) = 8.0000 day; remaining BOD (Lt) = 5.9000 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

   $$List the givens: rate constant (k) = 0.2400 1/day, time (t) = 8.0000 day, remaining BOD (Lt) = 5.9000 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 40.2437\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 40.2437 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

• 80.4873 — kept a factor of two that cancels in the correct rearrangement.
• 20.1218 — dropped that same factor in the other direction.
• 44.2680 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 140.0 mg/L; time (t) = 4.0000 day; remaining BOD (Lt) = 112.9 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

   $$List the givens: ultimate BOD (L_{0}) = 140.0 mg/L, time (t) = 4.0000 day, remaining BOD (Lt) = 112.9 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.0538\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.0538 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

• 0.1076 — kept a factor of two that cancels in the correct rearrangement.
• 0.0269 — dropped that same factor in the other direction.
• 0.0592 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 355.0 mg/L; rate constant (k) = 0.2300 1/day; time (t) = 4.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 355.0 mg/L, rate constant (k) = 0.2300 1/day, time (t) = 4.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 141.5\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 141.5 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

• 282.9 — kept a factor of two that cancels in the correct rearrangement.
• 70.7371 — dropped that same factor in the other direction.
• 155.6 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Test Solution and Seeding Procedures