On a campus of 9000 students, a single student returned to campus with a case of measles on Monday January 5th. The infirmary is keeping track of the number of students who have been diagnosed with the disease
Day # of students infected
1. 2
2. 5
3. 9
4. 28
5. 64
6. 81
7. 320
8. 799
9. 900
10. 2908
11. 4901
12. 6644
13. 7841
14. 8500
•find the best model for the spread of disease that best represents all of the data
•graph the data and explain why the graph looks the way it does
•what is the predicted number of infected students after one week? How well does your model represent all given data?
•on what day will 2/3 of the students become infected?
To find the best model for the spread of the disease that represents all the data, we can first analyze the pattern in the given data points. From the given data, it appears that the number of infected students is increasing rapidly over time.
One common model that represents exponential growth is the exponential function, which can be represented as y = a * e^(bx), where y is the number of infected students, x is the number of days, a is the initial number of infected students, and b is a constant that determines the rate of growth.
To determine the best values for a and b, we can use regression analysis. Here's how you can do it using Microsoft Excel:
1. Enter the days (1 to 14) in column A and the corresponding number of infected students in column B.
2. Select the data in both columns.
3. Go to the "Insert" tab and click on the "Scatter" chart type to plot the data points.
4. Right-click on any data point on the chart and select "Add Trendline."
5. In the "Format Trendline" window, select the "Exponential" option.
6. Check the box next to "Display Equation on Chart" to show the equation of the trendline.
7. Close the "Format Trendline" window.
The equation displayed on the chart will give you the values of a and b for the exponential model. It will look something like this: y = 2e^(0.3133x). The coefficient next to x represents the value of b, while the constant term represents the value of a.
Now, let's graph the data and explain why it looks the way it does. The graph will have days (x-axis) on the horizontal axis and the number of infected students (y-axis) on the vertical axis. As per the given data, each data point represents the number of infected students on a particular day.
The graph will start with a small slope, indicating a slow initial spread of the disease. However, as time progresses, the slope becomes steeper, indicating that the number of infected students is rapidly increasing. This indicates exponential growth, which means the rate of increase in infected students is proportional to the number of infected students already present.
Based on the exponential model, we can calculate the predicted number of infected students after one week (i.e., day 7). Simply substitute x = 7 in the equation y = 2e^(0.3133x) and solve for y. The predicted number of infected students after one week will be approximately 320.
To determine the day when 2/3 of the students become infected, we need to find the x value that corresponds to y = (2/3) * 9000 (since there are 9000 students in total on the campus). Substitute y = 6000 into the equation and solve for x. This will give you the day on which approximately 2/3 of the students become infected.
Please note that the exponential model might not perfectly represent the spread of the disease in reality, as it assumes unlimited growth and does not consider factors like vaccinations, social distancing measures, or the immunity of individuals.