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

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)`

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 = (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) = (1600)/(2000)`

`e^(4r) = 0.8`

`ln e^(4r) = ln 0.8`

`4r = -0.22`

`r = (-0.22)/(4)`

`r = -0.055`

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

`P(t) = 2000e^(-0.055t)`