SOLUTION: An unknown radioactive element decays into non-radioactive substances. In 560 days the radioactivity of a sample decreases by 51 percent. (a) What is the half-life of the eleme

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: An unknown radioactive element decays into non-radioactive substances. In 560 days the radioactivity of a sample decreases by 51 percent. (a) What is the half-life of the eleme      Log On


   

Question 1136372: An unknown radioactive element decays into non-radioactive substances. In 560 days the radioactivity of a sample decreases by 51 percent.
(a) What is the half-life of the element?
half-life:
(days)
(b) How long will it take for a sample of 100 mg to decay to 47 mg?
time needed:

Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
49=100%2Ae%5E%28-560k%29------first step is to find the value of k.
-
0.49=e%5E%28-560k%29
ln%280.49%29=-560k
k=-ln%280.49%29%2F560
k=0.0012738
MODEL: highlight_green%28y=pe%5E%28-0.0012738x%29%29

Note that the half life will be very close to 560 days.
More like 546 days.

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


The fraction of the material remaining after n half-lives is

%28.5%29%5E%28n%29

(a) Determine the half-life

In this problem, the amount of material decreased by 51% in 560 days; that means after 560 days 49% remains. Use the basic formula to determine how many half-lives that is.

%28.5%29%5En+=+0.49
n = 1.0291463 (to several decimal places, using a graphing calculator)

So the half-life of the material is

560%2F1.0291463+=+544.14

(b) Determine how long it will take a 100mg sample to decay to 47mg

Use the basic formula to determine the number of half-lives it takes for the material to decay to where 47% remains, then multiply the half-life by that number.

%28.5%29%5En+=+0.47
n+=+1.0892673
1.0892673%2A544.14+=+592.7139

It takes about 593 days for a 100mg sample to decay to 47mg.