Abacteria culture grows with constant relative growth rate. after 2 hours there are 600 bacteria and after 8 hours the count is 75,000.
(a) find the initial population.
(b) find an expression for the population after t hours.
(c) find the number of cells after 3 hours.
(d) find the rate of growth after 3 hours.
(e) when will the population reach 200,000?

a) The initial population is 125 bacteria.

b)

c) There are 1378 cells after 3 hours.

d) The rate of growth after 3 hours is of 1102 cells an hour.

e) The population will reach 200,000 in 9.22 hours.

Step-by-step explanation:

The population of this culture can be modeled by the following formula:

In which P(t) is the population after t hours, is the initial population, and r, as decimal, is the growth rate.

(a) Find the initial population.

Also

Replacing in this equation:

Applying ln to both sides

The initial population is:

The initial population is 125 bacteria.

(b) Find an expression for the population after t hours.

(c) Find the number of cells after 3 hours.

There are 1378 cells after 3 hours.

(d) Find the rate of growth after 3 hours.

This is P'(3).

So

The rate of growth after 3 hours is of 1102 cells an hour.

(e) When will the population reach 200,000?

Applying ln to both sides

The population will reach 200,000 in 9.22 hours.