Iodine-123 is a radioactive isotope used in medicine to test the function of the thyroid gland. Its half-life is 13.1 hours. A 500-microcurie (Ci) dose of iodine-123 is administered to a patient. The quantity, Q, left in the body after t hours is given by the equation Q=500(1/2)^(t/13.1). Answer parts (a) through (d).
a.)Determine the time it takes for the level of iodine-123 to drop to 65 μCi.
To determine the time it takes for the level of iodine-123 to drop to 65 μCi, we need to solve the equation Q = 65.
65 = 500(1/2)^(t/13.1)
Divide both sides by 500.
(1/2)^(t/13.1) = 65/500
Simplify.
(1/2)^(t/13.1) = 0.13
Take the logarithm of both sides with base 1/2.
log base 1/2 of 0.13 = t/13.1
Simplify.
log base 1/2 of 0.13 = t/13.1
Use the change of base formula to convert the logarithm to a common base.
log base 2 of 0.13 / log base 2 of (1/2) = t/13.1
Simplify.
log base 2 of 0.0842 = t/13.1
Use a calculator to find the value of the logarithm.
-3.5712 = t/13.1
Multiply both sides by 13.1.
-3.5712 * 13.1 = t
Solve for t.
t = -46.85692
Since time cannot be negative, we discard this solution.
Therefore, the time it takes for the level of iodine-123 to drop to 65 μCi is approximately 46.86 hours.