Questions: Assume simulitis is spreading through a population of 300 individuals. For this strain of simulitis, absent any constraints, each infected individual would cause another 2 infections per week, and the disease would thus initially spread at a rate of 200% per week. If the initial number of infections is given by P0=30, find the number of infections after one week and after two weeks. Round answers to the nearest whole number of cases, but only after completing the calculations. After 1 week, there are cases. After 2 weeks, there are cases.

Assume simulitis is spreading through a population of 300 individuals. For this strain of simulitis, absent any constraints, each infected individual would cause another 2 infections per week, and the disease would thus initially spread at a rate of 200% per week.

If the initial number of infections is given by P0=30, find the number of infections after one week and after two weeks. Round answers to the nearest whole number of cases, but only after completing the calculations.

After 1 week, there are cases.

After 2 weeks, there are cases.

Transcript text: Assume simulitis is spreading through a population of 300 individuals. For this strain of simulitis, absent any constraints, each infected individual would cause another 2 infections per week, and the disease would thus initially spread at a rate of $200 \%$ per week. If the initial number of infections is given by $P_{0}=30$, find the number of infections after one week and after two weeks. Round answers to the nearest whole number of cases, but only after completing the calculations. After 1 week, there are $\square$ cases. 1 After 2 weeks, there are $\square$ cases.

Solution

Solution Steps

To solve this problem, we need to model the spread of the infection using exponential growth. The number of infections after each week can be calculated using the formula for exponential growth: $ P(t) = P_0 \times (1 + r)^t $, where $ P_0 $ is the initial number of infections, $ r $ is the growth rate, and $ t $ is the time in weeks.

Calculate the number of infections after one week using the given growth rate.
Calculate the number of infections after two weeks using the same growth rate.

Step 1: Initial Parameters

We start with the initial number of infections given by $ P_0 = 30 $ and a growth rate of $ r = 2.0 $ (which corresponds to a $ 200\% $ increase per week).

Step 2: Calculate Infections After One Week

To find the number of infections after one week ($ t = 1 $), we use the formula for exponential growth: \[ P(1) = P_0 \times (1 + r)^1 \] Substituting the values: \[ P(1) = 30 \times (1 + 2) = 30 \times 3 = 90 \] Thus, after one week, the number of infections is $ P(1) = 90 $.

Step 3: Calculate Infections After Two Weeks

Next, we calculate the number of infections after two weeks ($ t = 2 $): \[ P(2) = P_0 \times (1 + r)^2 \] Substituting the values: \[ P(2) = 30 \times (1 + 2)^2 = 30 \times 3^2 = 30 \times 9 = 270 \] Thus, after two weeks, the number of infections is $ P(2) = 270 $.