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
1602
2014
3080
2004
1855
2016
3403
2006
2114
2018
3786
2008
2415
2020
4274
2010
2604
2022
4826
2012
2818
2024
5426
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 2030​?
​$
  
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.

Answer 1

To find a linear function that models the data provided for national expenditures for health care from years 2002 to 2024, we can follow these steps:

Part 1: Find the Linear Model

1. Define the variables:

   • Let \( x \) be the number of years after 2000 (e.g., for 2002, \( x = 2 \)).
   • Let \( y \) be the expenditures in billions of dollars.

2. Convert the data to points: The data points would be:

   • (2, 1602)
   • (4, 1855)
   • (6, 2114)
   • (8, 2415)
   • (10, 2604)
   • (12, 2818)
   • (14, 3080)
   • (16, 3403)
   • (18, 3786)
   • (20, 4274)
   • (22, 4826)
   • (24, 5426)

3. Calculate the slope (m): To find the slope \( m \), we can use two points. Let's take the first and last points, (2, 1602) and (24, 5426): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5426 - 1602}{24 - 2} = \frac{3824}{22} \approx 174.727 \]

4. Find the y-intercept (b): We can use the slope-intercept form \( y = mx + b \) and substitute one point to solve for \( b \). Using point (2, 1602): \[ 1602 = 174.727(2) + b \implies b \approx 1602 - 349.454 \approx 1252.546 \]

5. Create the linear equation: Thus the linear function can be expressed as: \[ y \approx 174.727x + 1252.546 \]

Final Form:

Rounding to three significant digits: \[ y \approx 174.727x + 1252.546 \approx 175x + 1253 \]

So the model is: \[ y = 175x + 1253 \]

Part 2: Predict Expenditures in 2030

To find the expenditure for the year 2030:

• \( x = 2030 - 2000 = 30 \)

Using the model: \[ y \approx 175(30) + 1253 \ y \approx 5250 + 1253 \ y \approx 6503 \]

Thus, the predicted expenditures will be approximately $6503 billion.

Part 3: Predicting $8069 Billion Expenditures

To find the year when the expenditures will be $8069 billion, set up the equation: \[ 8069 = 175x + 1253 \] Solving for \( x \): \[ 8069 - 1253 = 175x \ 6816 = 175x \ x \approx \frac{6816}{175} \approx 38.8 \]

Since \( x \) is the number of years after 2000, round \( 38.8 \) to the nearest whole number gives \( x = 39 \). Hence: \[ \text{Year} = 2000 + 39 = 2039 \]

Final Answers:

1. Linear Function: \( y = 175x + 1253 \)
2. Predicted expenditure in 2030: $6503 billion
3. Year when national expenditures reach $8069 billion: 2039