Question

Suppose that a Petri dish initially contains 2000 bacteria cells. An antibiotic is introduced and after 4 hour, there are now 1600 bacteria cells present. Let P(t) be the number of bacteria cells present t hours after the antibiotic is introduced. (a) (8 points) Suppose that P(t) is a linear function. Find a formula for P(t) (b) (8 points) Suppose that P(t) is an exponential function. Find a formula for P(t)

Answer 1

Answer:

a)The linear function for $P(t)$ is:

$P(t) = 2000 - 100*t$

b)The exponential function for $P(t)$ is:

$P(t) = 2000e^{-0.055t}$

Step-by-step explanation:

(a) Suppose that P(t) is a linear function. Find a formula for P(t):

$P(t)$ can be modeled by a linear function in the following format.

$P(t) = P_{0} - r*t$, in which $P_{0}$ 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, $P_{0} = 2000$

We have

$P(t) = 2000 - r*t$

An antibiotic is introduced and after 4 hour, there are now 1600 bacteria cells present. So $P(4) = 1600$. With this information, we can find the value of r.

$P(t) = 2000 - r*t$

$1600 = 2000 - r*(4)$

$4r = 400$

$r = \frac{400}{4}$

$r = 100$

So, the linear function for $P(t)$ is:

$P(t) = 2000 - 100*t$

b) Suppose that P(t) is an exponential function. Find a formula for P(t)

$P(t)$ can also be modeled by an exponential function in the following format:

$P(t) = P_{0}e^{rt}$

The values mean the same as in a). We use the fact that $P(4) = 1600$ to find r.

$P(t) = 2000e^{rt}$

$1600 = 2000e^{4r}$

$e^{4r} = \frac{1600}{2000}$

$e^{4r} = 0.8$

$ln e^{4r} = ln 0.8$

$4r = -0.22$

$r = \frac{-0.22}{4}$

$r = -0.055$

So, the exponential function for $P(t)$ is:

$P(t) = 2000e^{-0.055t}$