The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected. Source:Centers for Disease Control

Cheyanne Leigh 2020-11-08 Answered

The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected.
Source:
Centers for Disease Control and Prevention.
YearCases19952951997166199910920011152003942005107200779
a) Plot the data on a graphing calculator, letting t=0 correspond to the year 1995.
b) Using the regression feature on your calculator, find a quadratic, a cubic, and an exponential function that models this data.
c) Plot the three functions with the data on the same coordinate axes. Which function or functions best capture the behavior of the data over the years plotted?
d) Find the number of cases predicted by all three functions for 20152015. Which of these are realistic? Explain.

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Expert Answer

Nathaniel Kramer
Answered 2020-11-09 Author has 78 answers

Step 1
Answer for (a): The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected
YearCases19952951997166199910920011152003942005107200779
Now letting t=0 corresponds to the year 1995 we get the following table.
YeartCases19950295199721661999410920016115200389420051010720071279
Step 2
By using graphical calculator we have the following graph,
image
Step 3
Answer for (b) Using the regression feature on our calculator, a quadratic, a cubic, and an exponential function that models the above data are given as
y=2.3839t2  42.5536t + 269.2143
y= 0.4931t3 + 1.2589t2  81.998t + 292.881
y=213.7921(0.9149)t
Answer for(c) Let us plot above three function and data point on the same window we observe that the exponential function y=213.7921(0.9149)t is the best fit the data.
image
Step 4
Answer for (d) t=20 corresponds to the year 2015.
y(20)=2.383(20)2  42.5536(20) + 269.2143
=371.7023  372
y(20)= 0.4931(20)3 + 11.2589(20)2  81.998(20) + 292.881
= 788.319  788
y(20)=213.7921(0.9149)20
=36.0964  36
The only realistic value is given by the exponential function because the pattern of the data suggests that the number of cases decrease over time.

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