Contagious diseases can spread very quickly. Supposed five people are ill during the first week of an epidemic and that each person who is ill spreads the disease to four people by the end of the next week. By the end of the tenth week how many people have been affected by the illness?
I can't figure out how to set this up.
To determine the number of people affected by the illness by the end of the tenth week, we need to calculate the spread of the disease each week and add up the total number of affected people. Let's break it down step by step:
Week 1: We are given that five people are ill at the beginning of the epidemic, so that's our starting point.
Week 2: Each person who is ill during the first week spreads the disease to four people by the end of the second week. Therefore, the number of people ill in the second week is 5 * 4 = 20.
Week 3: Similarly, each person in the second week spreads the disease to four more people. Since there are 20 people ill in the second week, the number of people ill in the third week is 20 * 4 = 80.
Week 4: Following the same pattern, the number of people ill in the fourth week is 80 * 4 = 320.
Week 5: Continuing this process, the number of people ill in the fifth week is 320 * 4 = 1280.
Week 6: Similarly, the number of people ill in the sixth week is 1280 * 4 = 5120.
Week 7: The number of people ill in the seventh week is 5120 * 4 = 20480.
Week 8: The number of people ill in the eighth week is 20480 * 4 = 81920.
Week 9: The number of people ill in the ninth week is 81920 * 4 = 327680.
Week 10: Finally, the number of people ill in the tenth week is 327680 * 4 = 1,310,720.
Therefore, by the end of the tenth week, approximately 1,310,720 people will have been affected by the illness.
To figure out how many people have been affected by the illness by the end of the tenth week, we can use a geometric progression model.
In the first week, there are 5 people who are ill.
In the second week, each of the 5 people who are ill spreads the disease to 4 new people. So, the total number of people who are ill in the second week is 5 * 4 = 20.
In the third week, each of the 20 people who are ill spreads the disease to 4 new people. So, the total number of people who are ill in the third week is 20 * 4 = 80.
We can observe that each week the number of people who are ill is multiplied by 4.
Using this pattern, we can calculate the number of people who are ill by the end of the tenth week:
Illness in 1st week: 5 people
Illness in 2nd week: 5 * 4 = 20 people
Illness in 3rd week: 20 * 4 = 80 people
Illness in 4th week: 80 * 4 = 320 people
Continuing this pattern for 10 weeks, we can calculate:
Illness in 10th week: (4^9) * 5
Therefore, by the end of the tenth week, the number of people affected by the illness would be (4^9) * 5.