Answer:
a) ![\frac{dC}{dt} = rC](https://tex.z-dn.net/?f=%5Cfrac%7BdC%7D%7Bdt%7D%20%3D%20rC)
And for this case we can rewrite the model like this:
![\frac{dC}{C} = r dt](https://tex.z-dn.net/?f=%5Cfrac%7BdC%7D%7BC%7D%20%3D%20r%20dt)
If we integrate both sides we got:
![ln C = rt + k](https://tex.z-dn.net/?f=%20ln%20C%20%3D%20rt%20%2B%20k)
If we use exponentials for both sides we got:
![C = e^{rt} e^k = C_o e^{rt}](https://tex.z-dn.net/?f=%20C%20%3D%20e%5E%7Brt%7D%20e%5Ek%20%3D%20C_o%20e%5E%7Brt%7D)
For this case
and r = -0.16
So then our model would be given by:
![C(t) = 100 e^{-0.16 t}](https://tex.z-dn.net/?f=%20C%28t%29%20%3D%20100%20e%5E%7B-0.16%20t%7D)
Where t represent the number of hours
b) ![C(5) = 100 e^{-0.16*5}= 44.9329](https://tex.z-dn.net/?f=%20C%285%29%20%3D%20100%20e%5E%7B-0.16%2A5%7D%3D%2044.9329)
Step-by-step explanation:
Part a
For this case we can assume the proportional model given by:
![\frac{dC}{dt} = rC](https://tex.z-dn.net/?f=%5Cfrac%7BdC%7D%7Bdt%7D%20%3D%20rC)
And for this case we can rewrite the model like this:
![\frac{dC}{C} = r dt](https://tex.z-dn.net/?f=%5Cfrac%7BdC%7D%7BC%7D%20%3D%20r%20dt)
If we integrate both sides we got:
![ln C = rt + k](https://tex.z-dn.net/?f=%20ln%20C%20%3D%20rt%20%2B%20k)
If we use exponentials for both sides we got:
![C = e^{rt} e^k = C_o e^{rt}](https://tex.z-dn.net/?f=%20C%20%3D%20e%5E%7Brt%7D%20e%5Ek%20%3D%20C_o%20e%5E%7Brt%7D)
For this case
and r = -0.16
So then our model would be given by:
![C(t) = 100 e^{-0.16 t}](https://tex.z-dn.net/?f=%20C%28t%29%20%3D%20100%20e%5E%7B-0.16%20t%7D)
Where t represent the number of hours
Part b
For this case we can replace the value t=5 into the model and we got:
![C(5) = 100 e^{-0.16*5}= 44.9329](https://tex.z-dn.net/?f=%20C%285%29%20%3D%20100%20e%5E%7B-0.16%2A5%7D%3D%2044.9329)