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Answer:
A) The expression for the number of bacteria is .
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
where stands for the population of bacteria.
Writing as
, we get
.
Notice that this is a separable equation, so
.
Then, integrating in both sides of the equality:
.
We have,
.
Now, taking exponential
.
The next step is to find the value for the constant . We do this using the initial condition
. Recall that this is the initial population of bacteria. So,
.
Hence, the expression becomes
.
Now, we find the value for . We are going to use that
. Notice that
.
Then,
.
Taking logarithm
, so
.
So, , and approximating to the fourth decimal place we can take
. Hence,
.
B) To find the number of bacteria after 5 hours, we only need to evaluate the expression we have obtained in the previous exercise:
.
C) In this case we want to do the reverse operation: we want to find the value of t such that
.
This expression is equivalent to
.
Now, taking logarithm we have
.
Finally,
.
So, after 5.55 hours the population of bacteria will reach 30000.