Answer:
Differential equation:
,
,
,
.
Solution: ![A(t) = A_{o}\cdot e^{-\frac{t}{\tau} }](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A_%7Bo%7D%5Ccdot%20e%5E%7B-%5Cfrac%7Bt%7D%7B%5Ctau%7D%20%7D)
Step-by-step explanation:
Let suppose that rate of change of the amount of Carbon-14 atoms are directly proportional to the current amount of Carbon-14 atoms. That is:
![\frac{dA}{dt}\propto A(t)](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%5Cpropto%20A%28t%29)
![\frac{dA}{dt} = k\cdot A(t)](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%20%3D%20k%5Ccdot%20A%28t%29)
Where:
- Amount of Carbon-14 atoms, dimensionless.
- Proportionality constant, measured in
.
must be negative as death rate is constant and birth rate is zero. (
). Dimensionally, we can rewritte this constant as following:
![k = -\frac{1}{\tau}](https://tex.z-dn.net/?f=k%20%3D%20-%5Cfrac%7B1%7D%7B%5Ctau%7D)
Where
is the time constant (
), measured in years.
We can find the solution of the ordinary differential equation by separating each variable:
![\frac{dA}{dt} = -\frac{1}{\tau}\cdot A](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%20%3D%20-%5Cfrac%7B1%7D%7B%5Ctau%7D%5Ccdot%20A)
![\int {\frac{dA}{A} } = -\frac{1}{\tau} \int \, dt](https://tex.z-dn.net/?f=%5Cint%20%7B%5Cfrac%7BdA%7D%7BA%7D%20%7D%20%3D%20-%5Cfrac%7B1%7D%7B%5Ctau%7D%20%5Cint%20%5C%2C%20dt)
![\ln A(t) = -\frac{t}{\tau} + C](https://tex.z-dn.net/?f=%5Cln%20A%28t%29%20%3D%20-%5Cfrac%7Bt%7D%7B%5Ctau%7D%20%2B%20C)
![A(t) = e^{-\frac{t}{\tau}+C }](https://tex.z-dn.net/?f=A%28t%29%20%3D%20e%5E%7B-%5Cfrac%7Bt%7D%7B%5Ctau%7D%2BC%20%7D)
![A(t) = e^{C}\cdot e^{-\frac{t}{\tau} }](https://tex.z-dn.net/?f=A%28t%29%20%3D%20e%5E%7BC%7D%5Ccdot%20e%5E%7B-%5Cfrac%7Bt%7D%7B%5Ctau%7D%20%7D)
![A(t) = A_{o}\cdot e^{-\frac{t}{\tau} }](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A_%7Bo%7D%5Ccdot%20e%5E%7B-%5Cfrac%7Bt%7D%7B%5Ctau%7D%20%7D)
Where
is the initial amount of atoms of Carbon-14.
Answer:
i think that it is 8 and yeah