A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 385.0 mg/L; rate constant (k) = 0.2600 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

   $$List the givens: ultimate BOD (L_{0}) = 385.0 mg/L, rate constant (k) = 0.2600 1/day, time (t) = 7.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 62.3799\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 62.3799 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

• 124.8 — kept a factor of two that cancels in the correct rearrangement.
• 31.1900 — dropped that same factor in the other direction.
• 68.6179 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2200 1/day; time (t) = 6.0000 day; remaining BOD (Lt) = 191.8 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

   $$List the givens: rate constant (k) = 0.2200 1/day, time (t) = 6.0000 day, remaining BOD (Lt) = 191.8 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 718.0\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 718.0 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

• 1,436 — kept a factor of two that cancels in the correct rearrangement.
• 359.0 — dropped that same factor in the other direction.
• 789.8 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 110.0 mg/L; time (t) = 7.5000 day; remaining BOD (Lt) = 397.0 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

   $$List the givens: ultimate BOD (L_{0}) = 110.0 mg/L, time (t) = 7.5000 day, remaining BOD (Lt) = 397.0 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = -0.1711\ \text{1/day}$$

6. Step 6 — Check: returning k = -0.1711 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

• -0.3423 — kept a factor of two that cancels in the correct rearrangement.
• -0.0856 — dropped that same factor in the other direction.
• -0.1882 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 395.0 mg/L; rate constant (k) = 0.2200 1/day; time (t) = 9.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 395.0 mg/L, rate constant (k) = 0.2200 1/day, time (t) = 9.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 54.5373\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 54.5373 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

• 109.1 — kept a factor of two that cancels in the correct rearrangement.
• 27.2687 — dropped that same factor in the other direction.
• 59.9911 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.0900 1/day; time (t) = 5.0000 day; remaining BOD (Lt) = 375.3 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

   $$List the givens: rate constant (k) = 0.0900 1/day, time (t) = 5.0000 day, remaining BOD (Lt) = 375.3 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 588.6\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 588.6 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

• 1,177 — kept a factor of two that cancels in the correct rearrangement.
• 294.3 — dropped that same factor in the other direction.
• 647.4 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 310.0 mg/L; time (t) = 7.0000 day; remaining BOD (Lt) = 51.2000 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

   $$List the givens: ultimate BOD (L_{0}) = 310.0 mg/L, time (t) = 7.0000 day, remaining BOD (Lt) = 51.2000 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.2573\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.2573 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

• 0.5145 — kept a factor of two that cancels in the correct rearrangement.
• 0.1286 — dropped that same factor in the other direction.
• 0.2830 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 290.0 mg/L; rate constant (k) = 0.3000 1/day; time (t) = 6.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 290.0 mg/L, rate constant (k) = 0.3000 1/day, time (t) = 6.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 47.9367\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 47.9367 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

• 95.8734 — kept a factor of two that cancels in the correct rearrangement.
• 23.9683 — dropped that same factor in the other direction.
• 52.7303 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1500 1/day; time (t) = 5.5000 day; remaining BOD (Lt) = 223.3 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

   $$List the givens: rate constant (k) = 0.1500 1/day, time (t) = 5.5000 day, remaining BOD (Lt) = 223.3 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 509.5\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 509.5 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

• 1,019 — kept a factor of two that cancels in the correct rearrangement.
• 254.8 — dropped that same factor in the other direction.
• 560.5 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 120.0 mg/L; time (t) = 2.5000 day; remaining BOD (Lt) = 309.7 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

   $$List the givens: ultimate BOD (L_{0}) = 120.0 mg/L, time (t) = 2.5000 day, remaining BOD (Lt) = 309.7 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = -0.3792\ \text{1/day}$$

6. Step 6 — Check: returning k = -0.3792 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

• -0.7585 — kept a factor of two that cancels in the correct rearrangement.
• -0.1896 — dropped that same factor in the other direction.
• -0.4172 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 400.0 mg/L; rate constant (k) = 0.1300 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

   $$List the givens: ultimate BOD (L_{0}) = 400.0 mg/L, rate constant (k) = 0.1300 1/day, time (t) = 4.5000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 222.8\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 222.8 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

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

Reference: FE Reference Handbook — Environmental Engineering → National Research Council (NRC) Trickling Filter Performance