Find the half-life (in hours) of a radioactive substance that is reduced by 30 percent in 40 hours.
a = A e^-kt
.3 = e^-40 k
ln .7 = -.357 = -40 k
k = .008917
so
1/2 = e^-.008917 t if t is half life
ln .5 = - .693 = -.008917 t
t = 77.7 hours
typo in second line
.7 = e^-40 k
Why did the radioactive substance go to therapy? Because it had trouble with commitment! Anyway, let's get to the question.
To find the half-life, we need to figure out the time it takes for the substance to reduce by half. Since it reduces by 30% in 40 hours, we can assume it reduces by 50% over the half-life period.
So, if it reduces by 50% in the half-life, and it reduces by 30% in 40 hours, we can set up the following equation:
30% / 50% = 40 hours / half-life
Simplifying, we get:
0.3 / 0.5 = 40 hours / half-life
0.6 = 40 hours / half-life
Now let's solve for the half-life:
0.6 * half-life = 40 hours
half-life = 40 hours / 0.6
half-life ≈ 66.67 hours
So, the half-life of the radioactive substance is approximately 66.67 hours. Keep in mind that this answer may contain traces of comedy!
To find the half-life of a radioactive substance, we need to understand that the half-life is the amount of time it takes for the substance to reduce to half of its original value.
In this case, we are given that the substance is reduced by 30 percent in 40 hours. To determine the half-life, we need to find out how much time it takes for the substance to reduce to half of its original value, which is a reduction of 50 percent.
First, let's calculate the remaining percentage of the substance after the initial reduction. Since it is reduced by 30 percent, the remaining percentage is 100 percent minus 30 percent, which equals 70 percent.
Next, we need to find the time it takes for the substance to reduce from 70 percent to 50 percent. This is a reduction of 20 percent (70 percent - 50 percent), which represents half of the initial reduction.
To find the half-life, we can set up a proportion:
(initial reduction percentage / initial time) = (half reduction percentage / half-life)
In this case, the initial reduction percentage is 30 percent, the initial time is 40 hours, the half reduction percentage is 20 percent, and the half-life is what we are trying to find.
(30 percent / 40 hours) = (20 percent / half-life)
To solve for the half-life, we can cross-multiply:
(30 percent) * (half-life) = (40 hours) * (20 percent)
Simplifying the equation:
0.30 * half-life = 0.20 * 40
0.30 * half-life = 8
Divide both sides of the equation by 0.30 to solve for the half-life:
half-life = 8 / 0.30
Using a calculator, we find that the half-life is approximately 26.67 hours.