SOLUTION: A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Write an exponential equation f(t) representing this situat

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Write an exponential equation f(t) representing this situat      Log On


   

Question 1129640: A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams.
Write an exponential equation f(t) representing this situation. (Let f be the amount of radioactive substance in grams and t be the time in minutes.)
To the nearest minute, what is the half-life of this substance?
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
A scientist begins with 250 grams of a radioactive substance.
After 250 minutes, the sample has decayed to 32 grams.
Write an exponential equation f(t) representing this situation. (Let f be the amount of radioactive substance in grams and t be the time in minutes.)
To the nearest minute, what is the half-life of this substance?
:
A = Ao*2^(-t/h) is the radioactive decay formula, where
A = remaining amt after t time (32 gr)
Ao = initial amt (250 gr)
t = time (250 minutes)
h = half-life of substance
:
250*2^(-250/h) = 32
divide both sides by 250
2^(-250/h) = .128
ln(2^(250/h)) = ln(.128)
log equiv of exponent
-250%2Ft*ln(2) = ln(.128)
-250%2Ft = ln%28.128%29%2Fln%282%29
using your calc
-250%2Ft = -2.9658
-2.9658t = -250
t = %28-250%29%2F%28-2.9658%29
t = 84.3 ~ 84 minutes is the half life of the substance

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


I find the formula for radioactive decay easier to understand if we write it in terms of number of half-lives, instead of in terms of numbers of days (or hours, or years, or milliseconds).

The amount of a radioactive sample remaining after n half-lives is the original amount, 250g, multiplied by 1/2 to the power n.

f%28n%29+=+250%28.5%29%5En

We are given that after 250 minutes the amount remaining is 32g. Use that to find the number of half-lives there are in 250 minutes.

32+=+250%28.5%29%5En
0.128+=+.5%5En
log%28%28.128%29%29+=+n%2Alog%28%28.5%29%29
n+=+log%28%28.128%29%29%2Flog%28%28.5%29%29 = 2.9658 to 4 decimal places

The half life is then

250%2F2.9658 = 84.2947 minutes to 4 decimal places

The formula for the remaining amount as a function of the number of minutes t (with n = t/84.2947) is then

f%28t%29+=+250%28.5%29%5E%28t%2F84.2947%29