Exponential Growth and Decay
Find the half-life of a radioactive substance that decays by 5% in 9 years.
x = xi e^-kt
.95 = e^-9 k
ln .95 = -9k
k = .005699
.5 = e^-.005699 t
ln .5 = - .005699 t
t = 121.6 years
To find the half-life of a radioactive substance that decays by a certain percentage over a certain time period, we can use the formula for exponential decay:
N(t) = N0 * (1 - r)^t
Where:
- N(t) represents the remaining amount of the substance at time t.
- N0 represents the initial amount of the substance.
- r represents the decay rate (expressed as a decimal).
- t represents the time period.
In this case, the substance decays by 5%, so the decay rate would be 0.05 (or 5%/100 = 0.05). The time period is 9 years.
We want to find the half-life, which is the time it takes for the substance to decay by half of its original amount (N0/2). So we can set up the equation:
N(t) = N0/2
N0 * (1 - 0.05)^9 = N0/2
Now we can solve for N0:
(1 - 0.05)^9 = 1/2
0.95^9 = 1/2
0.4877 ≈ 1/2
Now we can solve for t:
t = 9 years
Therefore, using the given decay rate, it takes approximately 9 years for the substance to decay by half (or to reach its half-life).