Answer:
a)The linear function for
is:
![P(t) = 2000 - 100*t](https://tex.z-dn.net/?f=P%28t%29%20%3D%202000%20-%20100%2At)
b)The exponential function for
is:
![P(t) = 2000e^{-0.055t}](https://tex.z-dn.net/?f=P%28t%29%20%3D%202000e%5E%7B-0.055t%7D)
Step-by-step explanation:
(a) Suppose that P(t) is a linear function. Find a formula for P(t):
can be modeled by a linear function in the following format.
, in which
is the initial number of bacteria cells in the dish, t is the time and r is the rate that the number decreases.
Since the dish initially contains 2000 bacteria cells, ![P_{0} = 2000](https://tex.z-dn.net/?f=P_%7B0%7D%20%3D%202000)
We have
![P(t) = 2000 - r*t](https://tex.z-dn.net/?f=P%28t%29%20%3D%202000%20-%20r%2At)
An antibiotic is introduced and after 4 hour, there are now 1600 bacteria cells present. So
. With this information, we can find the value of r.
![P(t) = 2000 - r*t](https://tex.z-dn.net/?f=P%28t%29%20%3D%202000%20-%20r%2At)
![1600 = 2000 - r*(4)](https://tex.z-dn.net/?f=1600%20%3D%202000%20-%20r%2A%284%29)
![4r = 400](https://tex.z-dn.net/?f=4r%20%3D%20400)
![r = \frac{400}{4}](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B400%7D%7B4%7D)
![r = 100](https://tex.z-dn.net/?f=r%20%3D%20100)
So, the linear function for
is:
![P(t) = 2000 - 100*t](https://tex.z-dn.net/?f=P%28t%29%20%3D%202000%20-%20100%2At)
b) Suppose that P(t) is an exponential function. Find a formula for P(t)
can also be modeled by an exponential function in the following format:
![P(t) = P_{0}e^{rt}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P_%7B0%7De%5E%7Brt%7D)
The values mean the same as in a). We use the fact that
to find r.
![P(t) = 2000e^{rt}](https://tex.z-dn.net/?f=P%28t%29%20%3D%202000e%5E%7Brt%7D)
![1600 = 2000e^{4r}](https://tex.z-dn.net/?f=1600%20%3D%202000e%5E%7B4r%7D)
![e^{4r} = \frac{1600}{2000}](https://tex.z-dn.net/?f=e%5E%7B4r%7D%20%3D%20%5Cfrac%7B1600%7D%7B2000%7D)
![e^{4r} = 0.8](https://tex.z-dn.net/?f=e%5E%7B4r%7D%20%3D%200.8)
![ln e^{4r} = ln 0.8](https://tex.z-dn.net/?f=ln%20e%5E%7B4r%7D%20%3D%20ln%200.8)
![4r = -0.22](https://tex.z-dn.net/?f=4r%20%3D%20-0.22)
![r = \frac{-0.22}{4}](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B-0.22%7D%7B4%7D)
![r = -0.055](https://tex.z-dn.net/?f=r%20%3D%20-0.055)
So, the exponential function for
is:
![P(t) = 2000e^{-0.055t}](https://tex.z-dn.net/?f=P%28t%29%20%3D%202000e%5E%7B-0.055t%7D)