Question

A infection originated from an unknown pollution can kill 20 percent of those who are already ill and make 80 percent of the healthy population ill each month. Assume that everyone is healthy initially and the death rate of the healthy population is negligible. Suppose that no one who is infected can become healthy again. Find the percentage of the original population that will die in the next four months.

Answer 1

Approximately 43% of the original population will die in the next four months due to the infection, based on the given monthly infection and mortality rates.

Step-by-step explanation:

To calculate the percentage of the original population that will die in the next four months, we consider the infection rates and mortality rates provided. Initially, everyone is healthy, and we are given that each month, 80 percent of the remaining healthy population becomes ill. From the ill population, 20 percent dies each month. We can break this problem down into a month-by-month calculation to find the cumulative effect over four months.

Let's use the original population as 100% for simplicity. In the first month, 80% becomes ill, and 0% of the healthy population dies (since no death rate for the healthy was provided, only for those already ill). So, after the first month, 20% remains healthy, and 80% is ill.

In the second month, 80% of the remaining healthy population (which is now 20% of the original population) becomes ill. This is 16% of the original population, meaning 4% remains healthy. From the previously ill 80%, 20% dies, which is 16% of the original population. Thus, remaining ill are 64% of the original population.

For the third month, we again take 80% of the remaining healthy 4%, which is 3.2%, becoming ill, leaving 0.8% healthy. The ill population is now 64% + 3.2%, and 20% of the total ill population dies (67.2% * 20% = 13.44%).

In the fourth and final month analyzed, 80% of the remaining healthy 0.8% (which is 0.64%) becomes ill, and 20% of the now ill 67.84% population dies, which is 13.568% of the original population.

Ignoring the small fractions for ease of calculations, the approximate total death percentage over four months is the sum of the monthly death percentages: 16% (month 1) + 13.44% (month 3) + 13.568% (month 4). This equals around 43%, but the precise number would require exact fractional calculations. So, approximately 43% of the original population will have died due to the infection after four months.