Answer to Question #205356 in Differential Equations for Kate Bishop

Question

A certain plutonium isotope decays at a rate proportional to the amount present. Approximately 15% of the original amount decomposes in 100 years. How much amount of the substance has decayed after 600 years?

Answer 1

Exponential Decay Model

The initial value problem for exponential decay

$\dfrac{dA}{dt}=-kA, k>0, m(0)=m_0$

has particular solution

$A(t)=A_0e^{-kt},$

where $A_0=$original amount at time $t=0,$

$k=$relative decay rate that is constant $(k>0),$

$t=$ the time the isotope decays.

$A(t)=$ the amount that is left after time $t$ in years.

Given

$(1-0.15)A_0=A_0e^{-k(100)}$

Then

$e^{-k(100)}=0.85$

$100k=-\ln(0.85)$

$k=-0.01\ln(0.85)$

$A(600)=A_0e^{-k(600)}$

$A(600)=A_0(0.85)^6$

$\Delta A=A_0-A(600)$

$\Delta A\approx A_0-A_0(0.37715)$

$\Delta A\approx (0.62285)$

$62.285\%$ of the original amount has decayed after 600 years.