SOLUTION: There are 5500 lbs of a radioactive element with a half life of 5 years. What is its continuous decay rate? Please write your answer as a percentage using 2 decimal places.
Question 1167473: There are 5500 lbs of a radioactive element with a half life of 5 years. What is its continuous decay rate? Please write your answer as a percentage using 2 decimal places. Found 2 solutions by Theo, MathTherapy:Answer by Theo(13342) (Show Source): You can put this solution on YOUR website! the continuous growth / decay formula is f = p * e ^ (r * t)
f is the future value
p is the present value
e is the scientific constant of 2.718281828 rounded to 9 decimal places.
r is the growth / decay rate per time period.
t is the number of time periods.
in your problem, the time period is expressed in years.
to find the rate for the half line in 5 years, do the following:
1 = 2 * e ^ (r * 5)
divide both sides of this equation by 2 to get:
.5 = e ^ (r * 5)
take the natural log of both sides of this equation to get:
ln(.5) = ln(e ^ (r * 5))
since ln(e ^ (r * 5)) = r * 5 * ln(e), and since ln(e) = 1, the formula becomes:
ln(.5) = r * 5
divide both sides of this formula by 5 to get:
ln(.5) / 5 = r
solve for r to get:
r = -.1386294361.
confirm this rate is good by replacing r in the original equation with it and solving for f.
you get:
f = e ^ (-.1386294361 * 5)
solve for f to get:
f = .5
this confirms the value of r is good.
your solution is that the continuous decay rate is equal to -.1386294361 per year.
Answer by MathTherapy(10557) (Show Source): You can put this solution on YOUR website!
There are 5500 lbs of a radioactive element with a half life of 5 years. What is its continuous decay rate? Please write your answer as a percentage using 2 decimal places.
If ½ life is “a” years, then k, or Continuous Decay Rate, or DECAY CONSTANT = . In this case the continuous decay rate, or decay constant =
Note that the rate is negative (< 0), and should be since the element DECAYED over time.