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.