SOLUTION: An unknown radioactive element decays into non-radioactive substances. In 700 days the radioactivity of a sample decreases by 29 percent. (a) What is the half-life of the ele

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Question 1143472: An unknown radioactive element decays into non-radioactive substances. In
700 days the radioactivity of a sample decreases by 29 percent.
(a) What is the half-life of the element?
half-life:
(days)
(b) How long will it take for a sample of
100mg to decay to 54mg?
time needed:
(days)
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
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In 700 days the radioactivity of a sample decreases by 29 percent.
----------------------------------------------
??

y=pe%5E%28-kx%29
-
ln%28y%29=ln%28p%29-kx
ln%28y%29-ln%28p%29=-kx
kx=ln%28p%29-ln%28y%29
k=%28ln%28p%29-ln%28y%29%29%2Fx
k=%28ln%28p%2Fy%29%29%2Fx
-
k=ln%28100%2F71%29%2F700
k=0.0004893
-
highlight_green%28y=pe%5E%28-0.0004893x%29%29



HALF-LIFE
x%5Bhalf%5D=ln%282%29%2Fk
x%5Bhalf%5D=ln%282%29%2F0.0004893
1416

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


The radioactivity does not decrease. What you mean is the amount of radioactive material decreases by 29%.

(a) calculating the half life....

(1) Determine the number of half lives required for the amount of radioactive material to decrease by 29% -- i.e., to decay to 71% of the original amount.

%281%2F2%29%5En+=+0.71
n%2Alog%28%281%2F2%29%29+=+log%28%280.71%29%29
n+=+log%28%280.71%29%29%2Flog%28%281%2F2%29%29 = 0.4941 to 4 decimal places

(2) Determine the half life, given that 700 days is 0.4941 half lives.

half+life+=+700%2F0.4941 = 1416.69 days to 2 decimal places.

(a) ANSWER: the half life is 1416.69 days

(b) Determining the number of days for a sample of 100mg to decay to 54mg....

(1) Determine the number of half lives.
%281%2F2%29%5En+=+0.54%29
n%2Alog%28%281%2F2%29%29+=+log%28%280.54%29%29
n+=+log%28%280.54%29%29%2Flog%28%281%2F2%29%29 = 0.889 to 3 decimal places

(2) Determine the number of days in 0.889 half lives.
0.889%2A1416.69 = 1259 to the nearest whole number

(b) ANSWER: about 1259 days for 100mg to decay to 54mg.