Question

A patient is given a dose of iodine, Upper I dash 131, to treat an illness. The half-life of Upper I dash 131 is 8.04 days.



Approximately what percent of Upper I dash 131 is remaining on the 4th day?

Answer options with 4 options
A.
90%

B.
75%

C.
25%

D.
50%

Answer 1

To determine the percentage of \(^{131}\text{I}\) remaining after 4 days, we can use the half-life formula. The expression for the remaining quantity after a certain time is given by:

\[
N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]

Where:
- \(N(t)\) is the remaining quantity after time \(t\).
- \(N_0\) is the initial quantity.
- \(T_{1/2}\) is the half-life of the substance.
- \(t\) is the elapsed time.

Given:
- \(T_{1/2} = 8.04\) days
- \(t = 4\) days

Now, we can calculate how many half-lives have elapsed in 4 days:

\[
\text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{4 \text{ days}}{8.04 \text{ days}} \approx 0.4975
\]

Next, we can use this to find the remaining percentage:

\[
N(t) = N_0 \left( \frac{1}{2} \right)^{0.4975}
\]

Calculating \(\left( \frac{1}{2} \right)^{0.4975}\):

\[
\left( \frac{1}{2} \right)^{0.4975} \approx 0.7071 \quad (\text{using a calculator})
\]

To find the percentage remaining:

\[
\text{Percentage remaining} = 0.7071 \times 100\% \approx 70.71\%
\]

Since this value is approximately 71%, it does not exactly match any of the available options, but the closest approximation would be 75%.

Therefore, the answer is:

**B. 75%**