A population of bacteria is initially 1800 cells. After 2 hours, the population has increased to 1900 cells. Assume that the population is an exponential function of time. How many hours after the start will the instantaneous growth rate of the population be 200 cells per hour

The instantaneous growth rate of the bacteria will be 200 cells per hour after 3 hours.

Explanation:

There are four phases of bacterial growth represented by a bacterial growth curve, and they are; the lag phase, the log/exponential phase, the stationary phase, and the decline phase.

Under favorable conditions of nutrients, temperature, pH, oxygen and space, bacterial cells grow at a geometric or exponential rate this is the log phase, this means that the population doubles after a regular interval of time. In order to determine the time to attain a growth rate of 200 cells per hour, we have to start from the growth rate at 1 hour from the start and work upwards, and this is calculated thus;

it was given that after 2 hours, the population increased from 1800 to 1900, meaning that after 2 hours the growth rate was 100 cells per hour

∴ 0 to 2 hours = 100 cells/hour

2 hours = 100 cells/hour

∴ 1 hour = 100/2 = 50 cells/hour.

After the first hour, the growth rate was 50 cells/hour. Remembering that exponential growth is doubling of a population at an equal time interval (1 hour)

1 hour = 50 cells/hour

2 hours = 50 + 50 = 100 cells/hour

3 hours = 100 + 100 = 200 cells / hour

4 hours = 200 + 200 = 400 cells/hour

This continues until the bacteria reaches the stationary phase of the growth curve then begins to decline.

so from the calculation above, at 3 hours the growth rate is 200 cells per hour.

I suggest you read up bacterial growth phases for better understanding.