SOLUTION: Please help me solve this question. An initial 65 mg sample of a radioactive substance decays to 30 mg in 100 years. (a) Find the rate of decay of the substance. (b) Find the

Click here to see ALL problems on Exponential-and-logarithmic-functions

Question 1048828: Please help me solve this question.
An initial 65 mg sample of a radioactive substance decays to 30 mg in
100 years.
(a) Find the rate of decay of the substance.
(b) Find the half-life of the substance.
(c) How long would it take until only 10 mg of the amount remains?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the formula to use is f = p * e^(n * r)

f is the future value.
p is the present value
n is the number of time periods.
r is the interest rate per time period

p is equal to 65
f is equal to 30
n is equal to 100 years.
r would be the interest rate per year.

formula becomes 30 = 65 * e^(100*r).
divide both sides of this equation by 65 to get 30/65 = e^(100*r)
take the natural log of both sides of this equation to get ln(30/65) = ln(e^(100*r))
since ln(e^x) = x*ln(e) and since ln(e) = 1, this equation becomes ln(30/65) = ln(e^(100*r)) which becomes ln(30/65) = 100*r*ln(e) which becomes ln(30/65) = 100*r.
divide both sides of this equation by 100 and you get ln(30/65)/100 = r
solve for r to get r = ln(30/65)/100 = -.0077318989

that's your answer.
you can confirm it's true by replacing r in the original equation with -.0077318989 and the equation will be true.
you will get 30 = 65 * (e^(100*-.0077318989) which will then becomes 30 = 30 which is true.