The cumulative number of AIDS cases C in the United States from 1982 to 1994 can be modeled by C = 3200x2 + 1586, and the cumulative number of AIDS deaths D from 1982 to 1994 can be modeled by D = 1900x2 + 619. In these equations x = 0 corresponds to 1982, x = 1 corresponds to 1983, and so on until x = 12 corresponds

to 1994.* (Source: U.S. Department of Health.)
(a) Write the rational expression DC
in terms of x.
(b) Evaluate your expression for x = 4, 7, and 10.
Round your answers to the nearest thousandth.
Interpret the results.
(c) Explain what the rational expression DC
represents

(a) To write the rational expression DC in terms of x, we need to divide the cumulative number of AIDS deaths D by the cumulative number of AIDS cases C.

DC = D/C

(b) To evaluate the expression for x = 4, 7, and 10, we need to substitute these values into the rational expression DC.

For x = 4:
DC = D(4)/C(4) = (1900(4)^2 + 619)/(3200(4)^2 + 1586)
Evaluate the numerator and the denominator separately and then divide them:
Numerator = 1900(4)^2 + 619 = 30419
Denominator = 3200(4)^2 + 1586 = 52058
DC = 30419/52058 ≈ 0.584

For x = 7:
DC = D(7)/C(7) = (1900(7)^2 + 619)/(3200(7)^2 + 1586)
Again, evaluate the numerator and denominator separately:
Numerator = 1900(7)^2 + 619 = 96439
Denominator = 3200(7)^2 + 1586 = 158446
DC = 96439/158446 ≈ 0.609

For x = 10:
DC = D(10)/C(10) = (1900(10)^2 + 619)/(3200(10)^2 + 1586)
Evaluate numerator and denominator:
Numerator = 1900(10)^2 + 619 = 190619
Denominator = 3200(10)^2 + 1586 = 3221586
DC = 190619/3221586 ≈ 0.059

Interpretation of the results:
When x = 4, DC is approximately 0.584. This means that for the year 1986, the ratio of cumulative AIDS deaths to cumulative AIDS cases was approximately 0.584.

When x = 7, DC is approximately 0.609. This means that for the year 1989, the ratio of cumulative AIDS deaths to cumulative AIDS cases was approximately 0.609.

When x = 10, DC is approximately 0.059. This means that for the year 1992, the ratio of cumulative AIDS deaths to cumulative AIDS cases was approximately 0.059.

(c) The rational expression DC represents the ratio of cumulative AIDS deaths to cumulative AIDS cases for the given years. It provides insights into the severity and impact of AIDS on the affected population over time.