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.

If you can express the amount as

y = a*2^(-t/n)

then the half-life is n years. That is, as t increases by n, the amount is multiplied by 1/2

ae^-.0856t = a*2^(-t/n)
divide by a and take logs
-.0856t = -t/n ln 2
divide by t

.0856 = ln2/n
n = ln2/.0856 = .6931/.0856 = 8.1

So, the half-life is 8.1 days

check: wikipedia says the half-life is 8.02 days

To find the half-life of a substance, we need to determine the amount of time it takes for the substance to decay to half of its initial amount. In this case, we can set up the equation as follows:

y = ae^(-0.0856t)

where:
y = the remaining amount of radioactive iodine
a = the initial amount of radioactive iodine
t = time in days

Since we want to find the half-life, we know that y will be equal to half of the initial amount, so we can substitute y with a/2 in the equation:

a/2 = ae^(-0.0856t)

Next, we can divide both sides of the equation by a to isolate e^(-0.0856t):

1/2 = e^(-0.0856t)

To eliminate e, we can take the natural logarithm (ln) of both sides:

ln(1/2) = ln(e^(-0.0856t))

Using the property ln(e^x) = x, we can simplify further:

ln(1/2) = -0.0856t

Now, we can solve for t:

t = (ln(1/2)) / (-0.0856)

Using a calculator, we find that ln(1/2) ≈ -0.693147 and -0.0856 ≈ -0.09. Thus, the equation becomes:

t = (-0.693147) / (-0.09)

Simplifying, we get:

t ≈ 7.7016

Rounded to the tenths place, the half-life of radioactive iodine is approximately 7.7 days.