A bacteria culture grows with a constant relative growth rate. After 2 hours there are 400 bacteria and after 8 hours the count is 50,000. (a) Find the initial population. P(0) = 80 )ãbacteria
(b) Find an expression for the population after t hours. r(t) = 180( 125(6 Plt) =180(125(2))-
(c) Find the number of cells after 7 hours. (Round your answer to the nearest integer.) P(7)=72.358- bacteria
(d) Find the rate of growth after 7 hours. (Round your answer to the nearest integer.) P(7) 2x bacteria/hour
(e) When will the population reach 200,000? (Round your answer to one decimal place.) hours

a) P(0) = 80

b)

c) 22,363 cells.

d) The rate of growth after 7 hours is of 18,000 bacteria per hour.

e) 9.7 hours.

Step-by-step explanation:

A bacteria culture grows with a constant relative growth rate.

This means that the population is given by:

In which P(0) is the initial population and r is the growth rate, as a decimal.

After 2 hours there are 400 bacteria and after 8 hours the count is 50,000.

This means that in 6 hours, the population went from 400 bacteria to 50,000 bacteria. We use this to find r. So

So

(a) Find the initial population. P(0)

We have that P(2) = 400. We use this to find P(0). So

So

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

(c) Find the number of cells after 7 hours.

This is P(7). So

22,363 cells.

(d) Find the rate of growth after 7 hours.

We have to find the derivative when t = 7. So

The rate of growth after 7 hours is of 18,000 bacteria per hour.

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

This is t for which . So

So 9.7 hours.