Question

A particular isotope has a half-life of 81 days. If you start with 1 kilogram of this isotope, how much will remain after 160 days? after 320 days?(Round to three decimal places as needed.)

Answer 1

Given:

A particular isotope has a​ half-life of 81 days.

Required:

If the starting amount is 1 kilogram of this​ isotope, find the remaining amount after 160 ​days, after 320 ​days.

Explanation:

The amount for exponential decay using the half line is given by the formula:

$A=P(\frac{1}{2})^{\frac{t}{h}}$

Where A =accumulated amount

P = initial amount

t = elapsed time

h = half time

We have P = 1

h = 81

Find A when t = 160

$\begin{gathered} A=1(\frac{1}{2})^{\frac{160}{81}} \\ A=(\frac{1}{2})^{1.972} \\ A=0.254899 \\ A\approx0.255 \end{gathered}$

Find A when t = 320

$\begin{gathered} A=1(\frac{1}{2})^{\frac{320}{81}} \\ A=(\frac{1}{2})^{3.9506} \\ A=0.064677 \\ A\approx0.065 \end{gathered}$

Final Answer:

The remaining amount after 160 days is approximately 0.255 kg

and after 320 days is approximately 0.065 kg.