Question

A radio active isotope decays according to the exponential decay equation where t is in days. Round to the thousandths place. For the half life: The half life is the solution (t) of the equation : a2=ae−7.571t a 2 = a e − 7.571 t

Answer 1

The decay function is
$a(t)=a_{0} e^{-7.571t}$
where
a₀ = initial mass
t = time, days

At half life, a(t) = a₀/2, therefore the time required to achieve half life is given by the equation
$\frac{a_{0}}{2} =a_{0} e^{-7.751t}\\ \frac{1}{2} =e^{-7.571t}$

Take natural log of each side.
ln(1/2) = -7.571 t

Divide each side by -7.571 to obtain
$t= \frac{ln(1/2)}{-7.571} =0.0916$

Answer: 0.0192 days (nearest thousandth)