Respuesta :
Answer:
a)The linear function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000 - 100*t[/tex]
b)The exponential function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000e^{-0.055t}[/tex]
Step-by-step explanation:
(a) Suppose that P(t) is a linear function. Find a formula for P(t):
[tex]P(t)[/tex] can be modeled by a linear function in the following format.
[tex]P(t) = P_{0} - r*t[/tex], in which [tex]P_{0}[/tex] 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, [tex]P_{0} = 2000[/tex]
We have
[tex]P(t) = 2000 - r*t[/tex]
An antibiotic is introduced and after 4 hour, there are now 1600 bacteria cells present. So [tex]P(4) = 1600[/tex]. With this information, we can find the value of r.
[tex]P(t) = 2000 - r*t[/tex]
[tex]1600 = 2000 - r*(4)[/tex]
[tex]4r = 400[/tex]
[tex]r = \frac{400}{4}[/tex]
[tex]r = 100[/tex]
So, the linear function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000 - 100*t[/tex]
b) Suppose that P(t) is an exponential function. Find a formula for P(t)
[tex]P(t)[/tex] can also be modeled by an exponential function in the following format:
[tex]P(t) = P_{0}e^{rt}[/tex]
The values mean the same as in a). We use the fact that [tex]P(4) = 1600[/tex] to find r.
[tex]P(t) = 2000e^{rt}[/tex]
[tex]1600 = 2000e^{4r}[/tex]
[tex]e^{4r} = \frac{1600}{2000}[/tex]
[tex]e^{4r} = 0.8[/tex]
[tex]ln e^{4r} = ln 0.8[/tex]
[tex]4r = -0.22[/tex]
[tex]r = \frac{-0.22}{4}[/tex]
[tex]r = -0.055[/tex]
So, the exponential function for [tex]P(t)[/tex] is:
[tex]P(t) = 2000e^{-0.055t}[/tex]
