SOLUTION: If the half life of some radioactive material is 8 hours, how long will it take for 420 grams of the substance to decay to 10grams?

Algebra ->  Customizable Word Problem Solvers  -> Travel -> SOLUTION: If the half life of some radioactive material is 8 hours, how long will it take for 420 grams of the substance to decay to 10grams?      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 22115: If the half life of some radioactive material is 8 hours, how long will it take for 420 grams of the substance to decay to 10grams?
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
+y+=+y%5B0%5D%2Ae%5E%28kt%29+, where y%5B0%5D=+420 grams.

Since the half life is 8 hours, this means that when t=8, y=+.5%2Ay%5B0%5D
.5%2Ay%5B0%5D+=+y%5B0%5D%2A+e%5E%28k%2A8%29

Divide both sides by y%5B0%5D:
%28.5%2Ay%5B0%5D%29%2F%28y%5B0%5D%29+=+%28y%5B0%5D%2A+e%5E%28k%2A8%29%29%2F%28y%5B0%5D%29
.5+=+e%5E%288k%29

Solve for k, by taking the ln of each side of the equation:
+ln%28.5%29+=+ln%28e%5E%288k%29%29=+8k

Divide both sides by 8
+%28ln%28.5%29%29%2F8+=+k+

k = -.0866433976 approximately (rounded to calculator accuracy). You may want to store the number in your calculator for future use.

Now, the question is, given y= 10, how long did it take? In other words, find t, using the value of k from the calculator--don't round it off!!
+10+=+420+e%5E%28kt%29+, solve for t.

Divide both sides by 420:
10%2F420+=+e%5E%28kt%29+
1%2F42+=+e%5E%28kt%29+

I strongly recommend that you NOT calculate 1/42, since if you do, you will be tempted to round off the decimal. Do NOT round off numbers that you will use later to calculate your final answer. Leave it in fractional form, which is EXACT!

Take the ln of each side:
ln%281%2F42%29+=+ln%28e%5E%28kt%29%29++=+kt+

To find t, just divide both sides by k:
+%28ln%281%2F42%29%29%2Fk+=+t, where k is the value that you previously calculated and may have stored in your calculator:
t= 43.1385 approximately, which rounds off to about 43 or 43.1 years.

I did not check this, and I do make errors. If you find an error in this, please let me know!!

R^2 at SCC