Bacterial populations exhibit exponential growth when conditions such as nutrient availability and temperature are favorable. In this phase, cells reproduce through binary fission, where each cell divides into two identical daughter cells. This process causes the population to double at regular intervals, resulting in a growth rate that is directly proportional to the current number of cells. As the population increases, the number of new cells formed during each generation also grows, creating a steep rise in total cell count over time.
Exponential growth is highly predictable under constant conditions, as the doubling time remains unchanged regardless of the starting population size. This property allows scientists to estimate how quickly bacterial cultures will expand in laboratory or industrial processes. For example, a small initial population can rapidly reach millions of cells within a few hours if nutrients remain abundant.
Understanding exponential growth is essential in many fields, including medicine, where it helps explain how infections can escalate quickly if left untreated. It is also critical in biotechnology applications that rely on bacterial cultures for production. By recognizing the factors that sustain or limit exponential growth, researchers can better manage bacterial populations and design effective strategies for control or cultivation.
Cell populations in bacterial cultures are often modeled as exponential functions. Here, P of t shows the population function at time t, C shows the initial population, and k is the growth constant.
The initial population at time zero is given; substituting this into the population function gives the value of C.
Now this C is substituted back into the population function, leaving one unknown parameter.
Dividing both sides by 50 and then applying the natural logarithm gives the value of k.
The population at 1.5 hours is known. Substituting known values into the equation gives the value of k.
This gives the population function with the known constants.
The objective is to find the number of bacteria and the rate of growth at 3 hours.
To find the rate of growth, the derivative of the population function is taken, which is equal to the growth constant multiplied by the population function itself.
First, to find the number of bacteria at 3 hours, substitute three into the population function. This gives an estimated cell count.
Then, substituting this value into the derivative of the population function gives the rate of growth.
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