To model this situation we are going to use the exponential decay function:
![f(t)=a(1-b)^t](https://tex.z-dn.net/?f=f%28t%29%3Da%281-b%29%5Et)
where
![f(t)](https://tex.z-dn.net/?f=f%28t%29)
is the final amount remaining after
![t](https://tex.z-dn.net/?f=t)
years of decay
![a](https://tex.z-dn.net/?f=a)
is the initial amount
![b](https://tex.z-dn.net/?f=b)
is the decay rate in decimal form
![t](https://tex.z-dn.net/?f=t)
is the time in years
For substance A:
Since we have 300 grams of the substance,
![a=300](https://tex.z-dn.net/?f=a%3D300)
. To convert the decay rate to decimal form, we are going to divide the rate by 100%:
![r= \frac{0.15}{100} =0.0015](https://tex.z-dn.net/?f=r%3D%20%5Cfrac%7B0.15%7D%7B100%7D%20%3D0.0015)
. Replacing the values in our function:
![f(t)=a(1-b)^t](https://tex.z-dn.net/?f=f%28t%29%3Da%281-b%29%5Et)
![f(t)=300(1-0.0015)^t](https://tex.z-dn.net/?f=f%28t%29%3D300%281-0.0015%29%5Et)
![f(t)=300(0.9985)^t](https://tex.z-dn.net/?f=f%28t%29%3D300%280.9985%29%5Et)
equation (1)
For substance B:
Since we have 500 grams of the substance,
![a=500](https://tex.z-dn.net/?f=a%3D500)
. To convert the decay rate to decimal form, we are going to divide the rate by 100%:
![r= \frac{0.37}{100} =0.0037](https://tex.z-dn.net/?f=r%3D%20%5Cfrac%7B0.37%7D%7B100%7D%20%3D0.0037)
. Replacing the values in our function:
![f(t)=a(1-b)^t](https://tex.z-dn.net/?f=f%28t%29%3Da%281-b%29%5Et)
![f(t)=500(1-0.0037)^t](https://tex.z-dn.net/?f=f%28t%29%3D500%281-0.0037%29%5Et)
![f(t)=500(0.9963)^t](https://tex.z-dn.net/?f=f%28t%29%3D500%280.9963%29%5Et)
equation (2)
Since they are trying to determine how many years it will be before the substances have an equal mass
![M](https://tex.z-dn.net/?f=M)
, we can replace
![f(t)](https://tex.z-dn.net/?f=f%28t%29)
with
![M](https://tex.z-dn.net/?f=M)
in both equations:
![M=300(0.9985)^t](https://tex.z-dn.net/?f=M%3D300%280.9985%29%5Et)
equation (1)
![M=500(0.9963)^t](https://tex.z-dn.net/?f=M%3D500%280.9963%29%5Et)
equation (2)
We can conclude that the system of equations that can be used to determine <span>how long it will be before the substances have an equal mass, </span>
![M](https://tex.z-dn.net/?f=M)
, is:
![\left \{ {{M=300(0.9985)^t} \atop {M=500(0.9963)^t}} \right.](https://tex.z-dn.net/?f=%20%5Cleft%20%5C%7B%20%7B%7BM%3D300%280.9985%29%5Et%7D%20%5Catop%20%7BM%3D500%280.9963%29%5Et%7D%7D%20%5Cright.%20%20)
Solving the system, we can show that it will take approximately 231.59 years for that to happen.