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%

1 answer

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%