A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 320.0 mg/L; rate constant (k) = 0.2300 1/day; time (t) = 3.5000 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

   $$List the givens: ultimate BOD (L_{0}) = 320.0 mg/L, rate constant (k) = 0.2300 1/day, time (t) = 3.5000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 143.1\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 143.1 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

• 286.1 — kept a factor of two that cancels in the correct rearrangement.
• 71.5341 — dropped that same factor in the other direction.
• 157.4 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2800 1/day; time (t) = 3.5000 day; remaining BOD (Lt) = 179.3 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

   $$List the givens: rate constant (k) = 0.2800 1/day, time (t) = 3.5000 day, remaining BOD (Lt) = 179.3 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 477.7\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 477.7 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

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

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 280.0 mg/L; time (t) = 6.5000 day; remaining BOD (Lt) = 57.3000 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

   $$List the givens: ultimate BOD (L_{0}) = 280.0 mg/L, time (t) = 6.5000 day, remaining BOD (Lt) = 57.3000 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.2441\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.2441 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

• 0.4882 — kept a factor of two that cancels in the correct rearrangement.
• 0.1220 — dropped that same factor in the other direction.
• 0.2685 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 295.0 mg/L; rate constant (k) = 0.2200 1/day; time (t) = 3.5000 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

   $$List the givens: ultimate BOD (L_{0}) = 295.0 mg/L, rate constant (k) = 0.2200 1/day, time (t) = 3.5000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 136.6\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 136.6 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

• 273.2 — kept a factor of two that cancels in the correct rearrangement.
• 68.2944 — dropped that same factor in the other direction.
• 150.2 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2100 1/day; time (t) = 1.5000 day; remaining BOD (Lt) = 93.8000 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

   $$List the givens: rate constant (k) = 0.2100 1/day, time (t) = 1.5000 day, remaining BOD (Lt) = 93.8000 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 128.5\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 128.5 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

• 257.1 — kept a factor of two that cancels in the correct rearrangement.
• 64.2652 — dropped that same factor in the other direction.
• 141.4 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 245.0 mg/L; time (t) = 5.5000 day; remaining BOD (Lt) = 61.2000 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

   $$List the givens: ultimate BOD (L_{0}) = 245.0 mg/L, time (t) = 5.5000 day, remaining BOD (Lt) = 61.2000 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.2522\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.2522 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

• 0.5044 — kept a factor of two that cancels in the correct rearrangement.
• 0.1261 — dropped that same factor in the other direction.
• 0.2774 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 380.0 mg/L; rate constant (k) = 0.2900 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

   $$List the givens: ultimate BOD (L_{0}) = 380.0 mg/L, rate constant (k) = 0.2900 1/day, time (t) = 9.5000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 24.1714\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 24.1714 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

• 48.3429 — kept a factor of two that cancels in the correct rearrangement.
• 12.0857 — dropped that same factor in the other direction.
• 26.5886 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1700 1/day; time (t) = 6.5000 day; remaining BOD (Lt) = 333.2 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

   $$List the givens: rate constant (k) = 0.1700 1/day, time (t) = 6.5000 day, remaining BOD (Lt) = 333.2 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 1006\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 1,006 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

• 2,012 — kept a factor of two that cancels in the correct rearrangement.
• 503.0 — dropped that same factor in the other direction.
• 1,107 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 105.0 mg/L; time (t) = 9.5000 day; remaining BOD (Lt) = 13.0000 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

   $$List the givens: ultimate BOD (L_{0}) = 105.0 mg/L, time (t) = 9.5000 day, remaining BOD (Lt) = 13.0000 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.2199\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.2199 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

• 0.4398 — kept a factor of two that cancels in the correct rearrangement.
• 0.1099 — dropped that same factor in the other direction.
• 0.2419 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 205.0 mg/L; rate constant (k) = 0.1600 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

   $$List the givens: ultimate BOD (L_{0}) = 205.0 mg/L, rate constant (k) = 0.1600 1/day, time (t) = 6.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 78.4930\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 78.4930 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

• 157.0 — kept a factor of two that cancels in the correct rearrangement.
• 39.2465 — dropped that same factor in the other direction.
• 86.3423 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Loading Total System ≤ 35 pounds BOD5/(acre-day)