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.)

Question

Complete parts ​(a) through ​(c) below.
Year
Expenditures
Year
Expenditures

2002
1601
2014
3081
2004
1855
2016
3403
2006
2113
2018
3786
2008
2414
2020
4274
2010
2604
2022
4826
2012
2817
2024
5425
Question content area bottom
Part 1
a. 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.

yequals
  
enter your response herexplus
  
enter your response here
Part 2
b. What does the reported model predict the expenditures will be in 2040​?
​$
  
enter your response here billion ​(Use the answer from a to find this​ answer.)
Part 3
c. During what year does the reported model predict that the national expenditures for health care will be ​$8069 ​billion?
The reported model predicts that the national expenditures for health care will be ​$8069 billion in the year
  
enter your response here.
​(Use the answer from a to find this​ answer.)

Answer 1

To tackle the problem, we need to start earning a linear function from the provided data. We'll denote \(x\) as the number of years after 2000 (for example, \(x=2\) for the year 2002), and \(y\) as the national expenditures for health care in billions of dollars.

From the data table, we have the following points:

• (2, 1601)
• (4, 1855)
• (6, 2113)
• (8, 2414)
• (10, 2604)
• (12, 2817)
• (14, 3081)
• (16, 3403)
• (18, 3786)
• (20, 4274)
• (22, 4826)
• (24, 5425)

Part 1: Find a linear function

To find the linear function of the form \( y = mx + b \), we need to determine the slope \( m \) and the y-intercept \( b \).

1. Calculate the slope \( m \): We can choose two points to calculate the slope. Let's use the first and the last data points (2, 1601) and (24, 5425).

   \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5425 - 1601}{24 - 2} = \frac{3824}{22} \approx 174.7273 \]

2. Calculate the y-intercept \( b \): We can use one of the points to find \( b \). Using the point (2, 1601):

   \[ 1601 = 174.7273(2) + b \] \[ b = 1601 - 349.4546 \approx 1251.5454 \]

Thus, the linear model can be written as:

\[ y = 174.7273x + 1251.5454 \]

For 3 significant digits, the function is approximately:

\[ y \approx 175x + 1250 \]

The answer for Part 1:

\[ y = 175x + 1250 \]

Part 2: Predicting expenditures in 2040

To predict the expenditures in 2040, we first need the value of \( x \) for that year:

\[ x = 2040 - 2000 = 40 \]

Now, substitute \( x = 40 \) into the model:

\[ y \approx 175(40) + 1250 = 7000 + 1250 = 8250 \]

The answer for Part 2:

\[ \text{The model predicts that expenditures will be } $8250 \text{ billion.} \]

Part 3: Predicting when expenditures will be $8069 billion

We set the equation equal to $8069 billion:

\[ 8069 = 175x + 1250 \]

Now solve for \( x \):

\[ 8069 - 1250 = 175x \] \[ 6819 = 175x \] \[ x \approx \frac{6819}{175} \approx 38.66 \]

So, \( x \approx 39 \). Since \( x \) is years after 2000, the year can be calculated as:

\[ 2000 + 39 = 2039 \]

The answer for Part 3:

\[ \text{The reported model predicts that the national expenditures for health care will be } $8069 \text{ billion in the year } 2039. \]