A culture of the bacterium Salmonella enteritidis initially contains 50 cells. When introduced into a nutrient broth, the culture grows at a rate proportional to its size. After 1.5 hours, the population has increased to 775. (a) Find an expression for the number of bacteria after t hours. (Round your numeric values to four decimal places.)

The expression for the number of bacteria after t hours is

Step-by-step explanation:

When introduced into a nutrient broth, the culture grows at a rate proportional to its size.

This means that the size of the population, after t hours, is modeled by the following differential equation:

In which R is the growth rate.

The solution of this differential equation is:

In which P(0) is the initial population.

A culture of the bacterium Salmonella enteritidis initially contains 50 cells.

This means that , and so:

After 1.5 hours, the population has increased to 775.

This means that . We use this to find r. So

The expression is:

The expression for the number of bacteria after t hours is