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

`(a_(0))/(2) =a_(0) e^(-7.751t)\\ (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= (ln(1/2))/(-7.571) =0.0916`

Answer: 0.0192 days (nearest thousandth)