A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 210.0 mg/L; rate constant (k) = 0.1900 1/day; time (t) = 9.5000 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

   $$List the givens: ultimate BOD (L_{0}) = 210.0 mg/L, rate constant (k) = 0.1900 1/day, time (t) = 9.5000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 34.5396\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 34.5396 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

• 69.0793 — kept a factor of two that cancels in the correct rearrangement.
• 17.2698 — dropped that same factor in the other direction.
• 37.9936 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.0700 1/day; time (t) = 3.5000 day; remaining BOD (Lt) = 303.9 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

   $$List the givens: rate constant (k) = 0.0700 1/day, time (t) = 3.5000 day, remaining BOD (Lt) = 303.9 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 388.3\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 388.3 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

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

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 375.0 mg/L; time (t) = 7.5000 day; remaining BOD (Lt) = 371.3 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

   $$List the givens: ultimate BOD (L_{0}) = 375.0 mg/L, time (t) = 7.5000 day, remaining BOD (Lt) = 371.3 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.0013\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.0013 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

• 0.0026 — kept a factor of two that cancels in the correct rearrangement.
• 0.0007 — dropped that same factor in the other direction.
• 0.0015 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 260.0 mg/L; rate constant (k) = 0.0700 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

   $$List the givens: ultimate BOD (L_{0}) = 260.0 mg/L, rate constant (k) = 0.0700 1/day, time (t) = 4.5000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 189.7\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 189.7 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

• 379.5 — kept a factor of two that cancels in the correct rearrangement.
• 94.8726 — dropped that same factor in the other direction.
• 208.7 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1000 1/day; time (t) = 9.0000 day; remaining BOD (Lt) = 245.6 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

   $$List the givens: rate constant (k) = 0.1000 1/day, time (t) = 9.0000 day, remaining BOD (Lt) = 245.6 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 604.1\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 604.1 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

• 1,208 — kept a factor of two that cancels in the correct rearrangement.
• 302.0 — dropped that same factor in the other direction.
• 664.5 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 220.0 mg/L; time (t) = 3.0000 day; remaining BOD (Lt) = 188.3 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

   $$List the givens: ultimate BOD (L_{0}) = 220.0 mg/L, time (t) = 3.0000 day, remaining BOD (Lt) = 188.3 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.0519\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.0519 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

• 0.1037 — kept a factor of two that cancels in the correct rearrangement.
• 0.0259 — dropped that same factor in the other direction.
• 0.0571 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 355.0 mg/L; rate constant (k) = 0.0800 1/day; time (t) = 3.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 355.0 mg/L, rate constant (k) = 0.0800 1/day, time (t) = 3.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 279.3\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 279.3 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

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

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2100 1/day; time (t) = 2.0000 day; remaining BOD (Lt) = 256.5 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

   $$List the givens: rate constant (k) = 0.2100 1/day, time (t) = 2.0000 day, remaining BOD (Lt) = 256.5 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 390.4\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 390.4 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

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

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 320.0 mg/L; time (t) = 7.5000 day; remaining BOD (Lt) = 168.8 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

   $$List the givens: ultimate BOD (L_{0}) = 320.0 mg/L, time (t) = 7.5000 day, remaining BOD (Lt) = 168.8 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.0853\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.0853 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

• 0.1706 — kept a factor of two that cancels in the correct rearrangement.
• 0.0426 — dropped that same factor in the other direction.
• 0.0938 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 385.0 mg/L; rate constant (k) = 0.1800 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

   $$List the givens: ultimate BOD (L_{0}) = 385.0 mg/L, rate constant (k) = 0.1800 1/day, time (t) = 8.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 91.2172\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 91.2172 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

• 182.4 — kept a factor of two that cancels in the correct rearrangement.
• 45.6086 — dropped that same factor in the other direction.
• 100.3 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series