SOLUTION: A radioactive isotope has a half-life of 1.2 billion years. As measured by the presence of the isotope and its stable decay product, a rock originally contained 10 grams of the rad

Algebra ->  Exponents -> SOLUTION: A radioactive isotope has a half-life of 1.2 billion years. As measured by the presence of the isotope and its stable decay product, a rock originally contained 10 grams of the rad      Log On


   

Question 711861: A radioactive isotope has a half-life of 1.2 billion years. As measured by the presence of the isotope and its stable decay product, a rock originally contained 10 grams of the radioactive isotope, and now contains 1.25 grams. Approximately how many years old is the rock?
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Decay relationship is A=A%5B0%5De%5E%28kt%29, where here, t is billions of years.

FIND THE CONSTANT, k
half life is 1.2 billion years.
A=1%2F2, A%5B0%5D=1
1%2F2=1%2Ae%5E%28k%2A1.2%29
ln%281%2F2%29=ln%28e%5E%28k1.2%29%29
ln%281%2F2%29=1.2k%2A1
k=%28ln%281%2F2%29%29%2F1.2
k=-.578

HOW MUCH TIME PASSED or HOW OLD IS THE SAMPLE
highlight%28A=A%5B0%5De%5E%28-0.578t%29%29
Best to find t and then substitute values.
ln%28A%29=ln%28A%5B0%5D%2Ae%5E%28-0.578t%29%29
ln%28A%29=ln%28A%5B0%5D%29%2Bln%28e%5E%28-.578t%29%29
ln%28A%29=ln%28A%5B0%5D%29%2B-0.578%2At%2A1
ln%28A%29-ln%28A%5B0%5D%29=-0.578%2At
t=%28ln%28A%29-ln%28A%5B0%5D%29%29%2F%28-0.578%29
highlight%28t=%28ln%28A%5B0%5D%29-ln%28A%29%29%2F%280.578%29%29

Now substitute the given values to compute t.
highlight%28t=%28ln%2810%29-ln%281.25%29%29%2F0.578%29, billion years