Alleviating Mr. Miller’s Concerns Regarding Radioactive Iodine (I-131) Treatment

1. Remaining Iodine-131 After 6.8 Half-Lives

To calculate the amount of iodine-131 remaining after 6.8 half-lives, we use the formula:
[ \text{Amount remaining} = \text{Initial amount} \times (0.5)^{\text{number of half-lives}} ]

Given that the initial amount is 4 grams and the number of half-lives is 6.8:
[ \text{Amount remaining} = 4 \times (0.5)^{6.8} ]
[ \text{Amount remaining} = 4 \times (0.5)^{6.8} ≈ 0.04 ] grams

2. Time for Iodine-131 to go Through 3.3 Half-Lives

The time for iodine-131 to go through 3.3 half-lives can be calculated using the formula:
[ \text{Time} = \text{Number of half-lives} \times \text{Half-life} ]

Given that the number of half-lives is 3.3 and the half-life is 7.6 days:
[ \text{Time} = 3.3 \times 7.6 ≈ 24.68 ] days

3. Moles of Iodine-131 After 35 Days

To find the moles of iodine-131 left after 35 days when infused with 4.4 grams:
First, calculate the number of half-lives that have occurred:
[ \text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}} = \frac{35}{29} ≈ 1.2069 ]

Then, calculate the amount remaining using the formula:
[ \text{Amount remaining} = 4.4 \times (0.5)^{1.2069} ≈ 2.6071 ] grams
Convert grams to moles using the molar mass of iodine-131.

4. Time for Only 3.2 x 10^-2 grams to be Left

To find the time required for only 3.2 x 10^-2 grams to be left with an initial dose of 7.9 grams:
Calculate the number of half-lives needed to reach the final amount:
[ \text{Number of half-lives} = \log_2\left(\frac{\text{Final amount}}{\text{Initial amount}}\right) = \log_2\left(\frac{3.2 \times 10^{-2}}{7.9}\right) ≈ -4.5995 ]

Now, find the time taken for this by multiplying the number of half-lives by the half-life:
[ \text{Time} = -4.5995 \times 7.8 ≈ -35.92 ] days

In addressing Mr. Miller’s concerns, it is vital to provide reassurance based on accurate scientific calculations and explanations regarding the radioactive iodine treatment he is considering under Dr. Goodie’s care.

Let me know if you need further assistance or clarification on these calculations!