Question

the table shows the national expenditures for health care (in billions of dollars) for selected years from 2002 and projected to 2024. (these data include expenditures for medical research and medical facilities construction.)
complete parts (a) through (c) below.

( quad ) year ( quad ) expenditures ( quad ) year ( quad ) expenditures
( quad ) 2002 ( quad ) 1891 ( quad ) 2014 ( quad ) 3591
( quad ) 2004 ( quad ) 1995 ( quad ) 2016 ( quad ) 3953
( quad ) 2006 ( quad ) 2114 ( quad ) 2018 ( quad ) 4198
( quad ) 2008 ( quad ) 2414 ( quad ) 2020 ( quad ) 4714
( quad ) 2010 ( quad ) 2624 ( quad ) 2022 ( quad ) 4923
( quad ) 2012 ( quad ) 2818 ( quad ) 2024 ( quad ) 6425

1. find a linear function that models the data, with ( x ) equal to the number of years after 2000 and ( y ) equal to the expenditures for health care in billions of dollars. report the model with 3 significant digits.

( y = square x + square )

Explanation:

Step1: Define x and y values

Let \(x\) = years after 2000, so for 2002, \(x=2\), \(y=1901\); for 2024, \(x=24\), \(y=6026\) (using first and last data points for linear fit).

Step2: Calculate slope \(m\)

Slope formula: \(m=\frac{y_2-y_1}{x_2-x_1}\)
\(m=\frac{6026-1901}{24-2}=\frac{4125}{22}\approx187.5\)

Step3: Find y-intercept \(b\)

Use \(y=mx+b\) with \(x=2\), \(y=1901\):
\(1901=187.5(2)+b\)
\(1901=375+b\)
\(b=1901-375=1526\)

Step4: Write linear function

Round to 3 significant digits: \(m\approx188\), \(b\approx1530\)

Answer:

\(y = 188x + 1530\)