Question

In 1989, research scientists published a model for predicting the cumulative number of AIDS cases reported in the United States:

a(t) = 155((t − 1980)/10)^3 (thousands)

where t is the year. This paper was considered a "relief," since there was a fear the correct model would be of exponential type. Use the two data points predicted by the research model a(t) for the years 1989 and 1995 to construct a new exponential model b(t) for the number of cumulative AIDS cases. (Round values to three decimal places.)

I attemtped to solve this problem and got b(t) = 87.526e^(0.745)t but it was incorrect. How would I got about solving this problem?

Answer 1

Let us first find the two data points given to us. The previous mode is:

and the years are given as 1989 and 1995, so

Now, notice that the argument in the function a(t) has the term t - 1980, so, we can use something similar in our exponential model to make calculations easier, so we have, our exponential model as

Now, we have the values for t = 1989 and 1995, so using them we get

.......................(1)
........................(2)

Dividing (2) by (1) we get

Now, putting this in the first equation we get:

Thus, the required exponential model is: