SOLUTION: Last one tonight and I believe this will help complete my study guide for this chapter. Thank you in advance. The decay rate of a certain chemical is 9.5​% per year. What

Algebra ->  Probability-and-statistics -> SOLUTION: Last one tonight and I believe this will help complete my study guide for this chapter. Thank you in advance. The decay rate of a certain chemical is 9.5​% per year. What      Log On


   

Question 1130481: Last one tonight and I believe this will help complete my study guide for this chapter. Thank you in advance.
The decay rate of a certain chemical is 9.5​% per year. What is its ​half-life? Use the exponential decay model ​P(t)=P0e Superscript -kt)
−kt where k is the decay​ rate, and P0 is the original amount of chemical.
The​ half-life of the chemical is ___years.
​(Round to the nearest​ integer.)
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Model is P%28t%29=P%5Bo%5De%5E%28-kt%29, and half life is value of t when P%28t%29=%281%2F2%29%28P%5Bo%5D%29.

FIND k, first.
Some variables changes just for notational and keyboard conveniences.
y=p*e^(-kt) instead of P%28t%29 and P%5Bo%5D;
y=1%2Ae%5E%28-kt%29
1-0.095=1%2Ae%5E%28-k%2A1%29, one year of decay time
0.905=e%5E%28-k%29
ln%280.905%29=ln%28e%5E%28-k%29%29
-k=ln%280.905%29
-k=-0.099820
k=0.099820
Model can be stated highlight%28P%28t%29=P%5Bo%5De%5E%28-0.09982t%29%29.

FIND HALF-LIFE
Using model as 1%2F2=1%2Ae%5E%28-0.09982t%29
take natural logs of both sides, simplify, and solve for t.
result should be t%5B1%2F2%5D=ln%282%29%2F0.09982.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Last one tonight and I believe this will help complete my study guide for this chapter. Thank you in advance.
The decay rate of a certain chemical is 9.5​% per year. What is its ​half-life? Use the exponential decay model ​P(t)=P0e Superscript -kt)
−kt where k is the decay​ rate, and P0 is the original amount of chemical.
The​ half-life of the chemical is ___years.
​(Round to the nearest​ integer.)
matrix%281%2C3%2C+P%28t%29%2C+%22=%22%2C+P%5Bo%5De%5E%28-kt%29%29 

matrix%281%2C3%2C+P%28t%29%2FP%5Bo%5D%2C+%22=%22%2C+e%5E%28-+kt%29%29

matrix%281%2C3%2C+1%2F2%2C+%22=%22%2C+e%5E%28-+.095t%29%29 ------- Substituting matrix%281%2C7%2C+1%2F2%2C+for%2C+P%28t%29%2FP%5Bo%5D%2C+and%2C+.095%2C+for%2C+k%29

matrix%281%2C3%2C+-+.095t%2C+%22=%22%2C+ln+%281%2F2%29%29 ----- Converting to LOGARITHMIC (Natural) form