Radioactive iodine is used to determine the health of the thyroid gland. It decays according to the equation y=ae -0.0856t where t is in days. Find the half-life of this substance. Round to the tenths place.
To find the half-life of a substance, you need to determine the time it takes for half of the substance to decay. In this case, we can assume that "a" represents the initial amount of radioactive iodine.
The equation given is: y = ae^(-0.0856t)
Since we want to find the half-life, we can set y equal to half the initial amount (a/2) and solve for the corresponding value of t.
a/2 = ae^(-0.0856t)
To simplify, we can cancel out the common factor of "a" on both sides of the equation:
1/2 = e^(-0.0856t)
In order to solve for t, we need to isolate the exponential term. We can do this by taking the natural logarithm (ln) of both sides of the equation. Remember that the natural logarithm (ln) is the inverse function of the exponential function (e^x).
ln(1/2) = ln(e^(-0.0856t))
Using the property of logarithms that ln(e^x) = x:
ln(1/2) = -0.0856t
Now, we can solve for t by dividing both sides of the equation by -0.0856:
t = ln(1/2) / -0.0856
Using a calculator, we can evaluate this expression:
t ≈ 8.11 days
Therefore, the half-life of radioactive iodine in this scenario is approximately 8.1 days when rounded to the tenths place.