To determine how many half-value layers it would take to reduce an initial radiation intensity of [tex]\(4 \, \text{R}/\text{hr}\)[/tex] to a final intensity of approximately [tex]\(2 \, \text{mR}/\text{hr}\)[/tex], we need to follow a systematic approach using the concept of half-value layers.
Step-by-step Solution:
1. Understand the conversion:
- Initial intensity: [tex]\(4 \, \text{R}/\text{hr}\)[/tex]
- Final intensity: [tex]\(2 \, \text{mR}/\text{hr} = 0.002 \, \text{R}/\text{hr}\)[/tex]
(Note that [tex]\(1 \, \text{mR} = 0.001 \, \text{R}\)[/tex])
2. Formulate the relationship:
The relationship between the initial intensity [tex]\(I_0\)[/tex] and the final intensity [tex]\(I\)[/tex] after [tex]\(n\)[/tex] half-value layers can be described using the exponential decay formula:
[tex]\[
I = I_0 \times \left(\frac{1}{2}\right)^n
\][/tex]
where:
- [tex]\(I_0 = 4 \, \text{R}/\text{hr}\)[/tex]
- [tex]\(I = 0.002 \, \text{R}/\text{hr}\)[/tex]
3. Set up the equation and solve for [tex]\(n\)[/tex]:
[tex]\[
0.002 = 4 \times \left(\frac{1}{2}\right)^n
\][/tex]
To isolate [tex]\(n\)[/tex], we first divide both sides by 4:
[tex]\[
\left(\frac{1}{2}\right)^n = \frac{0.002}{4}
\][/tex]
Simplify the right-hand side:
[tex]\[
\left(\frac{1}{2}\right)^n = 0.0005
\][/tex]
4. Take the logarithm (base 2) of both sides:
[tex]\[
n = \log_2\left(0.0005\right)
\][/tex]
5. Calculate the value of [tex]\(n\)[/tex]:
Using the properties of logarithms:
[tex]\[
n \approx \log_2(2000)
\][/tex]
6. Logarithm value:
From the logarithm calculation, we know:
[tex]\[
\log_2(2000) \approx 10.965784284662087
\][/tex]
So, [tex]\(n \approx 10.97\)[/tex], which means approximately 11 half-value layers are required.
Final Answer:
D. 11
