Working with Real Data: Aids Cases

AIDS Cases: From 1993 to 2003 the cumulative number (N) of AIDS cases in thousands can be approximated by N = -2x2 + 76x + 430 where x = 0 corresponds to the year 1993.

Year 1993 1995 1997 1999 2001 2003

Cases 422 565 677 762 844 930

a) Use the equation to find N for each year in the table.

1993 (x = 0): N = -2(0)2 + 76(0) + 430

N = 0 + 0 + 430

N = 430

1995 (x = 2): N = -2(2)2 + 76(2) + 430

N = -8 + 152 + 430

N = 574

1997 (x = 4): N = -2(4)2 + 76(4) + 430

N = -32 + 304 + 430

N = 702

1999 (x = 6): N = -2(6)2 + 76(6) + 430

N = -72 + 456 + 430

N = 814

2001 (x = 8): N = -2(8)2 + 76(8) + 430

N = -128 + 608 + 430

N = 910

2003 (x = 10): N = -2(10)2 + 76(10) + 430

N = -200 + 760 + 430

N = 990

b) Discuss how well this equation approximates the data.

The approximated numbers generated by the equation N = -2x2 + 76x + 430 are close, but not exact matches, to the actual numbers of AIDS cases from 1993 t0 2003. In each computation, the difference between the actual number of cases and the number generated by the formula increased. In 1993 the difference was 1.86% and in 2003 the difference was 6.06%. The highest percentage difference was in 2001 when it was 7.25%.

c) Rewrite the equation with the right side completely factored.

N = -2x2 + 76x + 430

N = -2(x2 + 38x + 215)

N = -2(x – 43)(x + 5)

d) Use your equation from part (c) to find N for each year in the table. Do your answers agree with those found in part (a)?

1993 (x = 0): N = -2(0 - 43)(0 + 5)

N = -2(-43)(5)

N = 430

1995 (x = 2): N = -2(2 - 43)(2 + 5)

N = -2(-41)(7)

N = 574

1997 (x = 4): N = -2(4 - 43)(4 + 5)

N = -2(-39)(9)

N = 702

1999 (x = 6): N = -2(6 - 43)(6 + 5)

N = -2(-37)(11)

N = 814

2001 (x = 8): N = -2(8 - 43)(8 + 5)

N = -2(-35)(13)

N = 910

2003 (x = 10): N = -2(10 - 43)(10 + 5)

N = -2(-33)(15)

N = 990

All answers derived from the factored equation created...