The number of bacteria in a culture decreases according to a continuous exponential decay model. The initial population in a study is 3100 bacteria, and there are 868 bacteria left after 5 minutes.

(a) Let [tex]t[/tex] be the time (in minutes) since the beginning of the study, and let [tex]y[/tex] be the number of bacteria at time [tex]t[/tex]. Write a formula relating [tex]y[/tex] to [tex]t[/tex]. Use exact expressions to fill in the missing parts of the formula. Do not use approximations.

[tex]
y = 3100e^{kt}
[/tex]

(b) How many bacteria are there 17 minutes after the beginning of the study? Do not round any intermediate computations, and round your answer to the nearest whole number.

[tex]\square[/tex] bacteria



Answer :

Given that the number of bacteria in a culture decreases according to a continuous exponential decay model, we can solve the problem step-by-step as follows:

### Part (a)
We need to write a formula relating the number of bacteria [tex]\( y \)[/tex] at time [tex]\( t \)[/tex] in minutes.

The model for exponential decay can be expressed by the formula:

[tex]\[ y = y_0 \cdot e^{kt} \][/tex]

where:
- [tex]\( y_0 \)[/tex] is the initial population
- [tex]\( k \)[/tex] is the decay constant
- [tex]\( t \)[/tex] is the time elapsed

Given:
- [tex]\( y_0 = 3100 \)[/tex] (initial population)
- [tex]\( y = 868 \)[/tex] (population after 5 minutes)
- [tex]\( t = 5 \)[/tex] minutes

First, we need to determine the decay constant [tex]\( k \)[/tex]. We use the data for the population at 5 minutes to find [tex]\( k \)[/tex]:

[tex]\[ 868 = 3100 \cdot e^{5k} \][/tex]

Solving for [tex]\( k \)[/tex]:

[tex]\[ \frac{868}{3100} = e^{5k} \][/tex]

[tex]\[ e^{5k} = \frac{868}{3100} \][/tex]

[tex]\[ 5k = \ln\left(\frac{868}{3100}\right) \][/tex]

[tex]\[ k = \frac{1}{5} \ln\left(\frac{868}{3100}\right) \][/tex]

So the general formula for the number of bacteria at any time [tex]\( t \)[/tex] is:

[tex]\[ y = 3100 \cdot e^{\left(\frac{1}{5} \ln\left(\frac{868}{3100}\right) \cdot t \right)} \][/tex]

### Part (b)
We need to find the number of bacteria 17 minutes after the beginning of the study.

Using the formula we derived:

[tex]\[ y = 3100 \cdot e^{\left(\frac{1}{5} \ln\left(\frac{868}{3100}\right) \cdot 17 \right)} \][/tex]

To simplify the expression, remember that we already determined the decay constant [tex]\( k \)[/tex]:

[tex]\[ k = \frac{1}{5} \ln\left(\frac{868}{3100}\right) \approx -0.2545931351625775 \][/tex]

Then substituting [tex]\( k \)[/tex] and [tex]\( t = 17 \)[/tex] into the equation:

[tex]\[ y = 3100 \cdot e^{(-0.2545931351625775 \cdot 17)} \][/tex]

[tex]\[ y \approx 41 \][/tex]

Therefore, the number of bacteria 17 minutes after the beginning of the study is approximately 41 bacteria.