half-life of a radioactive isotope used in dose calculations Given decay constant (lambda) = 0.2015 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.2015 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.2015}$$

5. Step 5 — Evaluate:

   $$t_{half} = 3.4392\ \text{yr}$$

6. Step 6 — Check: returning t_half = 3.4392 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 6.8784 — kept a factor of two that cancels in the correct rearrangement.
• 1.7196 — dropped that same factor in the other direction.
• 3.7831 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life determination from a measured decay constant Given half-life (t_half) = 446.7 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 446.7 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0016\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0016 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0031 — kept a factor of two that cancels in the correct rearrangement.
• 0.0008 — dropped that same factor in the other direction.
• 0.0017 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

radioactive half-life used to plan waste storage duration Given decay constant (lambda) = 0.0114 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.0114 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.0114}$$

5. Step 5 — Evaluate:

   $$t_{half} = 60.7895\ \text{yr}$$

6. Step 6 — Check: returning t_half = 60.7895 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 121.6 — kept a factor of two that cancels in the correct rearrangement.
• 30.3947 — dropped that same factor in the other direction.
• 66.8684 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life of a radioactive isotope used in dose calculations Given half-life (t_half) = 4,803 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 4,803 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0001\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0001 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0003 — kept a factor of two that cancels in the correct rearrangement.
• 0.0001 — dropped that same factor in the other direction.
• 0.0002 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life determination from a measured decay constant Given decay constant (lambda) = 0.1921 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.1921 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.1921}$$

5. Step 5 — Evaluate:

   $$t_{half} = 3.6075\ \text{yr}$$

6. Step 6 — Check: returning t_half = 3.6075 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 7.2150 — kept a factor of two that cancels in the correct rearrangement.
• 1.8037 — dropped that same factor in the other direction.
• 3.9682 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

radioactive half-life used to plan waste storage duration Given half-life (t_half) = 5,919 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 5,919 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0001\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0001 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0002 — kept a factor of two that cancels in the correct rearrangement.
• 0.0001 — dropped that same factor in the other direction.
• 0.0001 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life of a radioactive isotope used in dose calculations Given decay constant (lambda) = 0.1075 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.1075 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.1075}$$

5. Step 5 — Evaluate:

   $$t_{half} = 6.4465\ \text{yr}$$

6. Step 6 — Check: returning t_half = 6.4465 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 12.8930 — kept a factor of two that cancels in the correct rearrangement.
• 3.2233 — dropped that same factor in the other direction.
• 7.0912 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life determination from a measured decay constant Given half-life (t_half) = 2,625 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 2,625 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0003\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0003 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0005 — kept a factor of two that cancels in the correct rearrangement.
• 0.0001 — dropped that same factor in the other direction.
• 0.0003 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

radioactive half-life used to plan waste storage duration Given decay constant (lambda) = 0.4411 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.4411 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.4411}$$

5. Step 5 — Evaluate:

   $$t_{half} = 1.5711\ \text{yr}$$

6. Step 6 — Check: returning t_half = 1.5711 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 3.1421 — kept a factor of two that cancels in the correct rearrangement.
• 0.7855 — dropped that same factor in the other direction.
• 1.7282 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life of a radioactive isotope used in dose calculations Given half-life (t_half) = 6,829 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 6,829 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0001\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0001 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0002 — kept a factor of two that cancels in the correct rearrangement.
• 0.0001 — dropped that same factor in the other direction.
• 0.0001 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life