Question

The pandemic is affecting the small village of Mathtown. On the first day, the mayor infected her administration clerk. The next day, they both infected someone new in the town. Each day, once someone was infected, they would infect another person in their community. How many days did it take to infect everyone in their town with a population of 2000? Express the final answer as a power. How many more days would it take if the population was twice as much - 4000?

Answer 1

It would take approximately 11 days to infect a town of 2000 people, and one additional day if the population was twice as much (4000 people). The final answers expressed as powers are 2^11 and 2^12, respectively.

Step-by-step explanation:

The problem is describing an exponential growth scenario where each infected person infects one other person each day. To solve how many days it would take to infect everyone in a town with a population of 2000, we start with the mayor as the first infected person on day 1. On day 2, there are 2 infected people and on day 3, there would be 4 infected people, and so on, doubling each day. The number of infected people can be expressed as 2n, where n is the number of days. We want to find n such that 2n ≥ 2000.

Using logarithms, we determine that it takes approximately 11 days to reach or surpass 2000 infected individuals, because 211 = 2048. Therefore, the answer expressed as a power is 211.

If the population were twice as much, 4000 people, it would take one more day for the entire population to be infected because the number of infected people would need to double once more, so 212 = 4096. Thus, the total would be 212, which is 12 days.