Answer:
A) The expression for the number of bacteria is [tex]P(t) = 400e^{0.7783t}[/tex].
B) After 5 hours there will be 19593 bacteria.
C) After 5.55 hours the population of bacteria will reach 30000.
Step-by-step explanation:
A) Here we have a problem with differential equations. Recall that we can interpret the rate of change of a magnitude as its derivative. So, as the rate change proportionally to the size of the population, we have
[tex]P' = kP[/tex]
where [tex]P[/tex] stands for the population of bacteria.
Writing [tex]P'[/tex] as [tex]\frac{dP}{dt}[/tex], we get
[tex]\frac{dP}{dt} = kP[/tex].
Notice that this is a separable equation, so
[tex]\frac{dP}{P} = kdt[/tex].
Then, integrating in both sides of the equality:
[tex]\int\frac{dP}{P} = \int kdt[/tex].
We have,
[tex]\ln P = kt+C[/tex].
Now, taking exponential
[tex]P(t) = Ce^{kt}[/tex].
The next step is to find the value for the constant [tex]C[/tex]. We do this using the initial condition [tex] P(0)=400[/tex]. Recall that this is the initial population of bacteria. So,
[tex]400 = P(0) = Ce^{k0}=C[/tex].
Hence, the expression becomes
[tex]P(t) = 400e^{kt}[/tex].
Now, we find the value for [tex]k[/tex]. We are going to use that [tex]P(4)=9000[/tex]. Notice that
[tex]9000 = 400e^{k4}[/tex].
Then,
[tex]\frac{90}{4} = e^{4k}[/tex].
Taking logarithm
[tex]\ln\frac{90}{4} = 4k[/tex], so [tex]\frac{1}{4}\ln\frac{90}{4} = k[/tex].
So, [tex]k=0.7783788273[/tex], and approximating to the fourth decimal place we can take [tex]k=0.7783[/tex]. Hence,
[tex]P(t) = 400e^{0.7783t}[/tex].
B) To find the number of bacteria after 5 hours, we only need to evaluate the expression we have obtained in the previous exercise:
[tex]P(5) =400e^{0.7783*5} = 19593.723 \approx 19593[/tex].
C) In this case we want to do the reverse operation: we want to find the value of t such that
[tex]30000 = 400e^{0.7783t}[/tex].
This expression is equivalent to
[tex]75 = e^{0.7783t}[/tex].
Now, taking logarithm we have
[tex]\ln 75 = 0.7783t[/tex].
Finally,
[tex]t = \frac{\ln 75}{0.7783} \approx 5.55[/tex].
So, after 5.55 hours the population of bacteria will reach 30000.
