SOLUTION: Suppose you are given a radioactive substance. After 30 days there is 1/10 of the original amount remaining. What is the half-life of the substance? I am confused here on what

Algebra ->  Functions -> SOLUTION: Suppose you are given a radioactive substance. After 30 days there is 1/10 of the original amount remaining. What is the half-life of the substance? I am confused here on what      Log On


   

Question 999529: Suppose you are given a radioactive substance. After 30 days there is 1/10 of the original amount remaining. What is the half-life of the substance?

I am confused here on what formula I should use and which is most efficient.
R(t) = re^(-kt)
where,
R(t) = the decay rate at time t
r = initial decay rate (at t=0)
t = time
k = the decay constant
Or
A = a(1/2)^(t/h)
Where,
A = final amount
a = initial amount
1/2 = split-factor
t = time
h = half-life
Please explain how to solve this problem using the one which you think is the most efficient.
Thank you
Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
The first function formula you show is a good choice, intended for continuous decay. Take logarithms of both sides and solve for k, and reuse some steps to find half-life.

For simpler work, use R to mean R(t).
R%28t%29+=+re%5E%28-kt%29
ln%28R%29=ln%28r%29-kt
ln%28R%29-ln%28r%29=-kt
kt=ln%28r%29-ln%28R%29
highlight_green%28k=ln%28r%2FR%29%2Ft%29

The description gave data for the one-tenth life, and you want to know half-life. Use the formula for k.

k=ln%2810%2F1%29%2F30
highlight_green%28k=69%29.


FIND HALF-LIFE

R=re%5E%28-kt%29, basic decay equation
R=re%5E%28-69t%29
but you have a step already which gives the formula you want to find half-life.

kt=ln%28r%2FR%29----from earlier steps
highlight_green%28t=ln%28r%2FR%29%2Fk%29
Values to use system%28r=1%2CR=1%2F2%2Ck=69%29
Now evaluate t for half-life.