Question 1129817: The half-life of Erbium-171 is 7.5 hours. What is the hourly decay rate? Express the result to four decimal places.
Express the hourly decay rate as a percentage to two decimal places.
Here is how I found the hourly decay rate for the first question:
ln(1/2)= ln(2)=ln(e^7.5)
-ln(2)=7.5k
K=- ln(2)/7.5
K=-0.0924
When I’ve gone to figure out the second portion it is incorrect.
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the half life is 7.5 hours.
what is the hourly decay rate?
the formula that uses continuous compounding to answer this would be:
f = p * e ^ (r * t), where ....
f = 1/2 and p = 1 and t = 7.5.
f is the future value.
p is the present value.
r is the interest rate per time period, which would be interest rate per hour.
t is the number of time periods, which would be number of hours.
the formula becomes:
1/2 = 1 * e ^ (r * 7.5)
this can be simplified to:
1/2 = e ^ (r * 7.5)
take the natural log of both sides of this equation to get:
ln(1/2) = ln(e ^ (r * 7.5)
ln(e ^ (r * 7.5) is the same as r * 7.5 * ln(e).
ln(e) is equal to 1.
the formula of ln(1/2) = ln(e ^ (r * 7.5) becomes:
ln(1/2) = r * 7.5.
solve for r to get:
r = ln(1/2) / 7.5.
this results in r = -.0924196241.
round this to 4 decimal places and the result is r = -.0924.
percent = rate * 100, therefore the percent would be r% = -9.24%.
if the first part is correct, then the second part is simply translating rate to percent.
if the rate is correct, then the percent has to be correct as well.
this assumes you used continuous compounding formula.
if you used discrete compounding formula, then the rate will not be the same.
the discrete compounding formula is f = p * (1 + r) ^ n
f is the future value, p is the present value, r is the interest rate per time period, n is the number of time periods.
the formula becomes 1/2 = 1 * (1 + r) ^ 7.5
this can be simplified to 1/2 = (1 + r) ^ 7.5
raise each side of this equation by the (1/7.5) power to get:
(1/2) ^ (1/7.5) = 1 + r.
subtract 1 from both sides of this equation to get:
(1/2) ^ (1/7.5) - 1 = r
solve for r to get:
r = -.0882775114.
i already confirmed this was correct.
round to 4 decimal places and r = -.0883.
convert to percent and percent r = -8.83%.
let me know if this helps you to get the correct answer and, if not, then what the correct answer needs to be, both in terms of rate and in terms of percent rate.
Answer by MathTherapy(10555)  (Show Source):
You can put this solution on YOUR website!
The half-life of Erbium-171 is 7.5 hours. What is the hourly decay rate? Express the result to four decimal places.
Express the hourly decay rate as a percentage to two decimal places.
Here is how I found the hourly decay rate for the first question:
ln(1/2)= ln(2)=ln(e^7.5)
-ln(2)=7.5k
K=- ln(2)/7.5
K=-0.0924
When I’ve gone to figure out the second portion it is incorrect.
You're CORRECT! k, or hourly decay rate is indeed: - .0924 (to 4 decimal places). Good job!
Now, all you have to do is MULTIPLY - .0924 by 100, which involves just moving the decimal point 2 places to the right of - .0924, to get: - 9.24%
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