In 1989, research scientists published a model for predicting the cumulative num
ID: 3038037 • Letter: I
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 1987 and 2000 to construct a new exponential model b(t) for the number of cumulative AIDS cases. (Round values to three decimal places.) b(t) = (thousands) Discuss how the two models differ and explain the use of the word "relief." This answer has not been graded yetExplanation / Answer
We havea(t) = 155[(t-1980)/10]3 ( in thousands), where t is the year and a(t) is the cumulative number of AIDS cases reported in US.
When t = 1987, we have a(1987) = 155[(1987-1980)/10]3 = 155(7/10)3= 155*0.343 = 53165 ( as the figures are in thousands). Also, when t = 2000, we have a(2000) = 155[(2000-1980)/10]3= 155(20/10)3 = 155*8 = 1240000(as the figures are in thousands).
A general form of an exponential function is pqt, where p, q are arbitrary real numbers. Let a(t) = pqt-1980 where t is the year. Since a(t) = 53165 in 1987, we have 53165 = pq(1987-1980) or, pq7 = 53165... (1) Similarly, pq20 = 1240000…(2) Hence, on dividing the 2nd equation by the 1st equation, we have (pq20)/(pq7) = 1240000/53165 or, q13 = 23.32361516 . Then q = (23.32361516)1/13 = 1.274 ( on rounding off to 3 decimal places). Then, from the 1st equation, we have p(1.274)7 = 53165 so that p = 53165/5.451 = 9753.256 (on rounding off to 3 decimal places). Then, the required exponential model is a(t) = (9753.256)(1.274)t-1980.
Before the research paper was published, the scientists in US expected the cumulative number of AIDS cases reported in US, to grow exponentially. However, the research paper allayed their fear as the formula reported in the research paper projected a much lower number of such cases. The formula came as a relief to the US scientists.