A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 160.0 mg/L; rate constant (k) = 0.2700 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

   $$List the givens: ultimate BOD (L_{0}) = 160.0 mg/L, rate constant (k) = 0.2700 1/day, time (t) = 7.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 24.1715\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 24.1715 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

• 48.3430 — 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 → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1700 1/day; time (t) = 3.5000 day; remaining BOD (Lt) = 301.8 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

   $$List the givens: rate constant (k) = 0.1700 1/day, time (t) = 3.5000 day, remaining BOD (Lt) = 301.8 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 547.2\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 547.2 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

• 1,094 — kept a factor of two that cancels in the correct rearrangement.
• 273.6 — dropped that same factor in the other direction.
• 601.9 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 300.0 mg/L; time (t) = 7.5000 day; remaining BOD (Lt) = 180.4 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

   $$List the givens: ultimate BOD (L_{0}) = 300.0 mg/L, time (t) = 7.5000 day, remaining BOD (Lt) = 180.4 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.0678\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.0678 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

• 0.1356 — kept a factor of two that cancels in the correct rearrangement.
• 0.0339 — dropped that same factor in the other direction.
• 0.0746 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 275.0 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

   $$List the givens: ultimate BOD (L_{0}) = 275.0 mg/L, rate constant (k) = 0.2600 1/day, time (t) = 7.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 44.5571\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 44.5571 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

• 89.1142 — kept a factor of two that cancels in the correct rearrangement.
• 22.2785 — dropped that same factor in the other direction.
• 49.0128 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.0500 1/day; time (t) = 7.5000 day; remaining BOD (Lt) = 279.7 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

   $$List the givens: rate constant (k) = 0.0500 1/day, time (t) = 7.5000 day, remaining BOD (Lt) = 279.7 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 407.0\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 407.0 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

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

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 275.0 mg/L; time (t) = 3.0000 day; remaining BOD (Lt) = 374.5 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

   $$List the givens: ultimate BOD (L_{0}) = 275.0 mg/L, time (t) = 3.0000 day, remaining BOD (Lt) = 374.5 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = -0.1029\ \text{1/day}$$

6. Step 6 — Check: returning k = -0.1029 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

• -0.2059 — kept a factor of two that cancels in the correct rearrangement.
• -0.0515 — dropped that same factor in the other direction.
• -0.1132 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 290.0 mg/L; rate constant (k) = 0.1800 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

   $$List the givens: ultimate BOD (L_{0}) = 290.0 mg/L, rate constant (k) = 0.1800 1/day, time (t) = 4.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 141.2\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 141.2 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

• 282.3 — kept a factor of two that cancels in the correct rearrangement.
• 70.5791 — dropped that same factor in the other direction.
• 155.3 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1200 1/day; time (t) = 7.0000 day; remaining BOD (Lt) = 12.4000 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

   $$List the givens: rate constant (k) = 0.1200 1/day, time (t) = 7.0000 day, remaining BOD (Lt) = 12.4000 mg/L$$

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

5. Step 5 — Evaluate:

   $$L_{0} = 28.7230\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 28.7230 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

• 57.4459 — kept a factor of two that cancels in the correct rearrangement.
• 14.3615 — dropped that same factor in the other direction.
• 31.5952 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 250.0 mg/L; time (t) = 9.0000 day; remaining BOD (Lt) = 97.2000 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

   $$List the givens: ultimate BOD (L_{0}) = 250.0 mg/L, time (t) = 9.0000 day, remaining BOD (Lt) = 97.2000 mg/L$$

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

5. Step 5 — Evaluate:

   $$k = 0.1050\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.1050 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

• 0.2099 — kept a factor of two that cancels in the correct rearrangement.
• 0.0525 — dropped that same factor in the other direction.
• 0.1155 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 380.0 mg/L; rate constant (k) = 0.0900 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

   $$List the givens: ultimate BOD (L_{0}) = 380.0 mg/L, rate constant (k) = 0.0900 1/day, time (t) = 10.0000 day$$

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

5. Step 5 — Evaluate:

   $$Lt = 154.5\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 154.5 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

• 309.0 — kept a factor of two that cancels in the correct rearrangement.
• 77.2482 — dropped that same factor in the other direction.
• 169.9 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → Daughter Product Activity