Abacteria culture starts with 400 bacteria and grows at a rate proportional to its size. after 4 hours, there are 9000 bacteria. a. find an expression for the number of bacteria after t hours. )t) b. find the number of bacteria after 5 hours. 19592 c. after how many hours will the population reach 3

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.

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