Question 1129817
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.