expoential and Logarithmic Models

Question

The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 1000 years?

Answer 1

more than 1/2 remains, more than 50, less than 100 mg

since 100 years is less than the half life of 1590 years

1/2 = e^1590r

ln.5 = 1590r

r = (ln.5)/1590 = rate of change = about -4.4 x 10^-4

P(t) = 100e^rt = 100e^100(ln.5)t/1590, solve for t = about

Answer 2

Quantity remaining = initial quantity * (1/2)^[(time in years)/(1590 years)]

Quantity remaining = 100 mg * (1/2)^(1000/1590)

Quantity remaining ≈ 64.6655474644 mg

Quantity remaining ≈ 64.7 mg, rounded to a more realistic accuracy.