SOLUTION: Find the half-life of a radioactive element, which decays according to the function A(t)= A 0A0e - 0.0291 te−0.0291t, where t is the time in years
Algebra.Com
Question 1138416: Find the half-life of a radioactive element, which decays according to the function A(t)= A 0A0e - 0.0291 te−0.0291t, where t is the time in years
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the formula for exponential growth or decay is f = p * e^(rt).
f is the future value
p is the present value
e is the scientific constant of 2.718281828.....
r is the rate per time period.
t is the number of time periods.
i can't figure out your formula very well, so it's hard to fit it into this formula.
you've got A(t)= A 0A0e - 0.0291 te−0.0291t, which i can translate somewhat into:
A(t) = (?????) * e^(-.0291 * t).
assuming you want to find the half life, you can make f = 1/2 and p = 1, so the formula becomes:
1/2 = e^(-.0291 * t)
take the natural log of both sides of this equation to get ln(1/2) = ln(e^(-.0291 * t)).
since ln(e^(-0.0291 * t) is equal to -.0291 * t * ln(e) and since ln(e) = 1, the formula becomes:
ln(1/2) = -.0291 * t
divide both sides of this equation by -.0291 and you get ln(1/2) / -.0291) = t
this results in t = 23.81949074.
the half life of this radioactive element should be equal to 23.81949074 years.
consider the original number being 2 and the future number being half that = 1.
the formula becomes 1 = 2 * e^(-.0291 * 23.81949074 years.) which becomes 1 = 1, confirming the solution is correct.
if your rate per year is .0291, whatever your present value is should be halved in 23.81949074 years.
this is because e^(-.0291 * 23.81949074 = .5, so any value of f = p * e^(-.0291 * 23.81949074) becomes f = p * .5 which means the value of p is halved when the formula is applied.
RELATED QUESTIONS
Find the half-life of a radioactive element that decays according to the rule:... (answered by stanbon)
If 250 mg of a radioactive element decays to 190 mg in 36hrs find the half-life of the... (answered by ikleyn)
Find the half-life of a certain radioactive substance which decays according to the... (answered by ikleyn)
Find the half-life of a certain radioactive substance which decays according to the... (answered by josgarithmetic,ikleyn)
Find the half-life of a certain radioactive substance which decays according to the... (answered by ikleyn)
Find the half-life of radium 226, which decays according to the function defined by A(t)... (answered by josmiceli)
A radioactive element decays at a rate of 7%. What is the half-life of the element (... (answered by ikleyn)
Radioactive strontium-90 is used in nuclear reactors and decays according to A =... (answered by ankor@dixie-net.com)
A certain radioactive element decays exponentially with a half-life of 5 hours. Find the... (answered by stanbon)