Final answer:
The initial population of bacteria approximately 4 hours before 7 pm, given a growth rate of 5% per hour and a later count of 1.012 x 10¹², is about 8.322 x 10¹¹.
Step-by-step explanation:
The question involves calculating the population of bacteria at a certain time given its growth rate and a later population count. The growth of curd clotting bacteria is 5% per hour, and we have a known population of 1.012 x 10¹² at 7 pm. To find the population 4 hours prior, we need to use the formula for exponential decay because we are effectively 'reversing' growth:
P = P0 * (1 + r)^-t
Where P is the final population, P0 is the initial population, r is the growth rate, and t is time. Rearranging the formula to solve for P0, we get:
P0 = P / (1 + r)^t
Substituting the given values:
P0 = (1.012 x 10¹²) / (1 + 0.05)^4
P0 = (1.012 x 10¹²) / (1.21550625)
P0 = Approximately 8.322 x 10¹¹ bacteria
Hence, the population of bacteria 4 hours before 7 pm was approximately 8.322 x 10¹¹.