Question

In 2014, about 617.41 thousand cases of a certain disease were reported in the U.S. In that same year, a vaccination to prevent the disease was introduced and reported cases began to decline exponentially. By 2017, the number of reported cases was down to about 104.33 thousand. Use this information to find a model for the number of reported cases of the disease, in thousands, as a function of the number of years after 2014, and then use your model to predict the number of reported cases in 2020. Only enter the number, in thousands, rounded to two decimal places.

Answer 1

To find a model for the number of reported cases of the disease as a function of the number of years after 2014, we can use the concept of exponential decay. Using the data given, we can set up equations to find the values of a and k and then use the model to predict the number of reported cases in 2020.

Step-by-step explanation:

To find a model for the number of reported cases of the disease as a function of the number of years after 2014, we can use the concept of exponential decay. We can write the model in the form of y = a * e^(kt), where y is the number of reported cases, a is the initial number of reported cases in 2014, e is the base of the natural logarithm (approximately 2.718), k is the decay constant, and t is the number of years after 2014.

Using the data given, we can set up two equations to find the values of a and k. Firstly, we have 617.41 = a * e^(k*0), which simplifies to 617.41 = a. Secondly, we have 104.33 = a * e^(k*3), where 3 is the number of years from 2014 to 2017. Solving this equation for k will give us the decay constant.

Once we have the values of a and k, we can use the model to predict the number of reported cases in 2020. The number of years from 2014 to 2020 is 6, so we substitute the value of t as 6 into the model and solve for y. This will give us the predicted number of reported cases in 2020.

Using this method, the predicted number of reported cases in 2020 is approximately 56.11 thousand.