Exponential Growth

Question

The half life of a certain substance is 27 years. How fast is a 500-gram sample of this substance decaying after 115 years?

 

Q=Q₀e^kt   and k=(ln(1/2)/half-life

Answer 1

Q(t) = Q₀e^kt  [note k must be negative to have decay]

 

dQ/dt = kQ₀e^kt

 

Find k from the data.

 

t_1/2 = 27 years ⇒

 

k = ln(1/2) / t_1/2 = ln(0.5) / 27 years^-1 ≅ -0.025672 years^-1

 

Q₀ = 500 g

 

Q(t) = (500 g)e^{-0.025672 t}, t in years

 

At t = 115 years,

 

dQ/dt = -0.025672(500 g) e^{-0.025672(115)} ≅ -0.67030 g/year