Let's solve the problem step-by-step while answering each part:
### Part (a):
We need to derive the formula relating [tex]\( y \)[/tex] (the number of bacteria) to [tex]\( t \)[/tex] (the time in minutes).
1. Initial Information:
- Initial population ([tex]\( y_0 \)[/tex]): [tex]\( 3100 \)[/tex] bacteria
- Remaining population after time [tex]\( t \)[/tex]): [tex]\( 868 \)[/tex] bacteria after [tex]\( 5 \)[/tex] minutes
2. Exponential Decay Formula:
We use the model [tex]\( y = y_0 \cdot e^{kt} \)[/tex], where [tex]\( k \)[/tex] is the decay constant.
3. Calculate the Decay Constant [tex]\( k \)[/tex]:
We know that [tex]\( y = 868 \)[/tex] when [tex]\( t = 5 \)[/tex].
So, we can plug these values into our formula:
[tex]\[
868 = 3100 \cdot e^{5k}
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
[tex]\[
\frac{868}{3100} = e^{5k}
\][/tex]
[tex]\[
e^{5k} = \frac{868}{3100}
\][/tex]
Taking the natural logarithm on both sides:
[tex]\[
5k = \ln \left( \frac{868}{3100} \right)
\][/tex]
[tex]\[
k = \frac{\ln \left( \frac{868}{3100} \right)}{5}
\][/tex]
5. Formula:
Substituting [tex]\( k \)[/tex] back into our exponential decay formula, we get:
[tex]\[
y = 3100 \cdot e^{\left( \frac{\ln \left( \frac{868}{3100} \right)}{5} \right) t}
\][/tex]
So the formula relating [tex]\( y \)[/tex] to [tex]\( t \)[/tex] is:
[tex]\[
y = 3100 \cdot e^{\left( \frac{\ln \left( \frac{868}{3100} \right)}{5} \right) t}
\][/tex]
### Part (b):
Now we need to find the number of bacteria after 17 minutes.
1. Given:
[tex]\[
t = 17 \text{ minutes}
\][/tex]
2. Using the Formula:
Plug [tex]\( t = 17 \)[/tex] into the formula we derived:
[tex]\[
y = 3100 \cdot e^{\left( \frac{\ln \left( \frac{868}{3100} \right)}{5} \times 17 \right)}
\][/tex]
3. Calculate the Number of Bacteria:
Evaluating this expression directly:
[tex]\[
y \approx 41
\][/tex]
So, after 17 minutes, the number of bacteria in the culture is approximately 41.
### Final Answer:
(a) The formula relating [tex]\( y \)[/tex] to [tex]\( t \)[/tex] is:
[tex]\[
y = 3100 \cdot e^{\left( \frac{\ln \left( \frac{868}{3100} \right)}{5} \right) t}
\][/tex]
(b) The number of bacteria after 17 minutes is:
[tex]\[
41 \text{ bacteria}
\][/tex]
