A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 58 cells. (a) Find the relative growth rate. (Assume t is measured in hours.) k = (b) Find an expression for the number of cells after t hours. P(t) = (c) Find the number of cells after 8 hours. cells (d) Find the rate of growth after 8 hours. (Round your answer to three decimal places.) billion cells per hour (e) When will the population reach 20,000 cells? (Round your answer to two decimal places.) hr

a) k=2.08 1/hour

b) The exponential growth model can be written as:

c) 977,435,644 cells

d) 2.033 billions cells per hour.

e) 2.81 hours.

Step-by-step explanation:

We have a model of exponential growth.

We know that the population duplicates every 20 minutes (t=0.33).

The initial population is P(t=0)=58.

The exponential growth model can be written as:

For t=0, we have:

If we use the duplication time, we have:

Then, we have the model as:

The relative growth rate (RGR) is defined, if P is the population and t the time, as:

In this case, the RGR is k=2.08 1/h.

After 8 hours, we will have:

The rate of growth can be calculated as dP/dt and is:

For t=8, the rate of growth is:

(2.033 billions cells per hour).

We can calculate when the population will reach 20,000 cells as:

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: