daughter product maximum activity time in a radioactive decay chain Given parent decay constant (lambda_1) = 0.1670 1/day; daughter decay constant (lambda_2) = 1.2370 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.1670 1/day, daughter decay constant (lambda_2) = 1.2370 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 1.8714\ \text{day}$$

6. Step 6 — Check: returning t_max = 1.8714 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.7429 — kept a factor of two that cancels in the correct rearrangement.
• 0.9357 — dropped that same factor in the other direction.
• 2.0586 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

maximum activity time of a daughter isotope following parent decay Given parent decay constant (lambda_1) = 0.0430 1/day; daughter decay constant (lambda_2) = 0.3720 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.0430 1/day, daughter decay constant (lambda_2) = 0.3720 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 6.5583\ \text{day}$$

6. Step 6 — Check: returning t_max = 6.5583 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 13.1167 — kept a factor of two that cancels in the correct rearrangement.
• 3.2792 — dropped that same factor in the other direction.
• 7.2142 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

daughter product activity peak time computed from decay constants Given parent decay constant (lambda_1) = 0.2150 1/day; daughter decay constant (lambda_2) = 0.3250 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.2150 1/day, daughter decay constant (lambda_2) = 0.3250 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 3.7562\ \text{day}$$

6. Step 6 — Check: returning t_max = 3.7562 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 7.5125 — kept a factor of two that cancels in the correct rearrangement.
• 1.8781 — dropped that same factor in the other direction.
• 4.1319 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

daughter product maximum activity time in a radioactive decay chain Given parent decay constant (lambda_1) = 0.1930 1/day; daughter decay constant (lambda_2) = 1.1050 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.1930 1/day, daughter decay constant (lambda_2) = 1.1050 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 1.9133\ \text{day}$$

6. Step 6 — Check: returning t_max = 1.9133 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.8266 — kept a factor of two that cancels in the correct rearrangement.
• 0.9566 — dropped that same factor in the other direction.
• 2.1046 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

maximum activity time of a daughter isotope following parent decay Given parent decay constant (lambda_1) = 0.0320 1/day; daughter decay constant (lambda_2) = 1.4620 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.0320 1/day, daughter decay constant (lambda_2) = 1.4620 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 2.6726\ \text{day}$$

6. Step 6 — Check: returning t_max = 2.6726 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 5.3452 — kept a factor of two that cancels in the correct rearrangement.
• 1.3363 — dropped that same factor in the other direction.
• 2.9399 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

daughter product activity peak time computed from decay constants Given parent decay constant (lambda_1) = 0.1400 1/day; daughter decay constant (lambda_2) = 1.6910 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.1400 1/day, daughter decay constant (lambda_2) = 1.6910 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 1.6063\ \text{day}$$

6. Step 6 — Check: returning t_max = 1.6063 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.2127 — kept a factor of two that cancels in the correct rearrangement.
• 0.8032 — dropped that same factor in the other direction.
• 1.7670 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

daughter product maximum activity time in a radioactive decay chain Given parent decay constant (lambda_1) = 0.2120 1/day; daughter decay constant (lambda_2) = 1.9010 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.2120 1/day, daughter decay constant (lambda_2) = 1.9010 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 1.2987\ \text{day}$$

6. Step 6 — Check: returning t_max = 1.2987 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.5975 — kept a factor of two that cancels in the correct rearrangement.
• 0.6494 — dropped that same factor in the other direction.
• 1.4286 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

maximum activity time of a daughter isotope following parent decay Given parent decay constant (lambda_1) = 0.1730 1/day; daughter decay constant (lambda_2) = 0.4900 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.1730 1/day, daughter decay constant (lambda_2) = 0.4900 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 3.2843\ \text{day}$$

6. Step 6 — Check: returning t_max = 3.2843 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 6.5685 — kept a factor of two that cancels in the correct rearrangement.
• 1.6421 — dropped that same factor in the other direction.
• 3.6127 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

daughter product activity peak time computed from decay constants Given parent decay constant (lambda_1) = 0.0140 1/day; daughter decay constant (lambda_2) = 0.5140 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.0140 1/day, daughter decay constant (lambda_2) = 0.5140 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 7.2063\ \text{day}$$

6. Step 6 — Check: returning t_max = 7.2063 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 14.4127 — kept a factor of two that cancels in the correct rearrangement.
• 3.6032 — dropped that same factor in the other direction.
• 7.9270 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time

daughter product maximum activity time in a radioactive decay chain Given parent decay constant (lambda_1) = 0.1680 1/day; daughter decay constant (lambda_2) = 1.5010 1/day, determine the time of maximum daughter activity (t_max) in day.

Start with the thinking

• The governing relation printed in this handbook section is Daughter Product Maximum Activity Time.
• Everything except t_max is given, so isolate t_max 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.
• The daughter product maximum activity time gives the elapsed time after which the daughter radionuclide activity reaches its peak.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

2. Step 2 — Rearrange symbolically for t_max:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

3. Step 3 — List the givens: parent decay constant (lambda_1) = 0.1680 1/day, daughter decay constant (lambda_2) = 1.5010 1/day.

4. Step 4 — Substitute the given values:

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}$$

5. Step 5 — Evaluate:

   $$t_{max} = 1.6429\ \text{day}$$

6. Step 6 — Check: returning t_max = 1.6429 day to

   $$t_{max} = \dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.2857 — kept a factor of two that cancels in the correct rearrangement.
• 0.8214 — dropped that same factor in the other direction.
• 1.8071 — rounded an intermediate value before the final step.

Reference: FE Handbook — Daughter Product Maximum Activity Time