Nuclear medicine technologists use the Iodine isotope I-131, with a half-life of 8 days,to check thyroid function of patients. After ingesting a tablet containing the iodine, the isotopes collect in a patient's thyroid, and a special camera is used to view its function. Suppose a person ingests a tablet contaiing 9 microcuries of I-131. To the nearest hour, how long will it be until there are only 2.8 microcuries in the person's thyroid?
the answers are given, it is 324 hours. im just not sure how to get that.
where did you get -6.93 and 24 from?
m(t)=9x2^(-t/192) 192=24x8
2.8=9x2^(-t/192)
t=192xln(9/2.8)/ln2
t=323.4236495
To determine the time it takes for the amount of I-131 in a person's thyroid to reach 2.8 microcuries, we can use the concept of half-life.
The half-life of I-131 is given as 8 days. This means that after every 8-day period, the amount of I-131 will reduce by half.
Let's calculate the number of half-lives required for the amount of I-131 to decrease from 9 microcuries to 2.8 microcuries:
Initial amount of I-131: 9 microcuries
Final amount of I-131: 2.8 microcuries
Half-life of I-131: 8 days
To find the number of half-lives, we can use the formula:
Number of half-lives = (log(final amount/initial amount)) / (log(0.5))
Number of half-lives = (log(2.8/9)) / (log(0.5))
Number of half-lives ≈ -0.903 / -0.301
Number of half-lives ≈ 3
From this calculation, we find that it takes approximately 3 half-lives for the amount of I-131 to decrease from 9 microcuries to 2.8 microcuries.
Since each half-life corresponds to 8 days, the total time can be calculated by multiplying the number of half-lives by the length of one half-life:
Total time = Number of half-lives × Length of one half-life
Total time = 3 × 8 days
Total time ≈ 24 days
Therefore, it will take approximately 24 days (to the nearest hour) for the amount of I-131 in the person's thyroid to decrease to 2.8 microcuries.
2.8=9*e^(-6.93 t/(8*24))
ln (2.8/9)=-6.93 t/182
solve for t in hours. I get about 31 hours