Final answer:
The size of a bacterial population after t hours can be calculated using the exponential growth formula, accounting for the doubling period of 50 minutes and an initial population of 4 bacteria. This results in a rapidly increasing population size and a J-shaped growth curve over time.
Step-by-step explanation:
To determine the size of the bacterial population after t hours, given that the initial population is 4 and it doubles every 50 minutes, we use the formula for exponential growth:
y(t) = P0 × 2(t × 60) / 50
Where P0 is the initial population (4 bacteria), t is the time in hours, and 60/50 converts hours into the 50-minute intervals in which the bacteria divide.
To find y(t), a specific value of t can be substituted into the formula to calculate the population at that time.
For example, after 3 hours (180 minutes), the number of generations, n, would be 180/50 = 3.6. The population size y(3) would be:
y(3) = 4 × 23.6 ≈ 4 × 13.49 ≈ 54 bacteria.
The size of the population increases dramatically due to exponential growth, following a pattern also visible in J-shaped growth curves.
The population of bacteria after t hours, starting with 4 bacteria and doubling every 50 minutes, is calculated using the exponential growth formula y(t) = 4 × 2(t × 60) / 50. The growth rate visibly accelerates, producing a J-shaped curve over time.