m₀ = 40 g, original mass
m₁ = 5.0 g, mass remaining after 30 days.
The decay equation is of the form
[tex]m(t) = m_{0} e^{-kt}[/tex]
where
t = time, days
k = constant
Therefore
[tex]40e^{-30k} = 5 \\ e^{-30k} = 5/40 = 0.125 \\ -30k = ln(0.125) \\ k = ln(0.125)/-30 = 0.0693[/tex]
At half-life, m = 20 g.
The time for half-life is
[tex]e^{-0.0693t} = 1/2 \\ -0.0693t = ln(0.5) \\ t = ln(0.5)/-0.0693 = 10 \, days[/tex]
Answer: 10 days
