The Law of Exponential Decay
The Law of exponential Decay is an application of antiderivatives. Just as something can decay, it can grow as well. Both decay and growth are included in the theorem below. This theorem also seems to show up on the Calc AP AB Test.
Exponential Growth and Decay
If y is a differentiable function of t such that y > 0 and y ' = ky, for some constant k, then y = Cekt.
C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay when k < 0.
Example:
Situation
10 grams of plutonium isotope Pu-239 is released in some far off nuclear research facility. They have exposed the 10 grams of Pu-239 to air, so it has turned yellow from its original silver like appearance. You're job is to find out how long it will take for the 10 grams of Pu-239 to decay to 1 gram of Pu-239. The researchers give you the fact that the half life of Pu-239 is 24,360
Solution: Let y represent the mass in grams of the plutonium. Because the rate of decay is proportional to y, you know that y = Cekt where t is the time in years. To find the values of the constants C and k, apply the initial conditions. Using the fact that y = 10 when t = 0, you can write 10 = Ce0 which implies that C = 10.
Use all known information
By using the half life fact that y = 5 when t = 24,360, you can write
5 = 10e24,360k
1/2 = e24,360k
(1/24,360)ln 1/2 = k
y = 10e-0.00028454t
Solve the obtained equation
Since you want to find the t when Pu-239 is 1, you set y = 1. Then solve the equation.
1 = 10e-0.00028454t
t = 80,922 years.
Click here to see the Pu-239 from the nuclear research facility. Notice that it is on blue screen.
