Question

The radioactive isotope radium has a half-life of 1,600 years. The mass of a 100-gram sample of radium will be______ grams after 200 years.

Answer 1

To solve this we are going to use the half life equation $N(t)=N_{0} e^{( \frac{-0.693t}{t _{1/2} }) }$
Where:
$N_{0}$ is the initial sample
$t$ is the time in years
$t_{1/2}$ is the half life of the substance
$N(t)$ is the remainder quantity after $t$ years 

From the problem we know that:
$N_{0} =100$
$t=200$
$t_{1/2} =1600$

Lets replace those values in our equation to find $N(t)$:
$N(200) =100e^{( \frac{(-0.693)(200)}{1600}) }$
$N(200)=100e^{( \frac{-138.6}{1600} )}$
$N(200)=100e^{-0.086625}$
$N(200)=91.7$

We can conclude that after 1600 years of radioactive decay, the mass of the 100-gram sample will be 91.7 grams.