People who are severely obese ​(BMIgreater than or equals​40) are at most risk for serious health​ problems, which are the most expensive to treat. The percent of people in a country who are severely obese from 2000 and projected to 2030 is shown in the table.

Question

Year
2000
2010
2015
2020
2025
2030

​ % Severely Obese
2.3
4.8
6.5
7.8
9.3
11.1
Complete parts ​(a) through ​(c) below.
Question content area bottom
Part 1
a. Find the linear function that models severe​ obesity, with x equal to the number of years after 2000 and y equal to the percent of severe obesity in the country.
yequals
  
0.293xplus
  
2.3
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 2
b. Graph the model and the data set on the same​ axes, and comment on the fit. Choose the correct answer below.
All graphs have viewing window​ [0,35] by​ [0,14] with Xsclequals5 and Ysclequals1.
A.

A coordinate system has a horizontal axis with tick marks dividing it into 7 regular intervals to the right of the vertical axis and has a vertical axis with tick marks dividing it into 14 intervals above the horizontal axis. The following points are plotted: (0, 2.3); (10, 4.8); (15, 6.5); (20, 7.8); (25, 9.3); (30, 11.1). A line rises from left to right and passes through the points (0, 2.1) and (30, 8.7). All plotted points are above the line. All values are approximate.
B.

A coordinate system has a horizontal axis with tick marks dividing it into 7 regular intervals to the right of the vertical axis and has a vertical axis with tick marks dividing it into 14 intervals above the horizontal axis. The following points are plotted: (0, 2.3); (5, 4.8); (10, 6.5); (15, 7.8); (20, 9.3); (25, 11.1). A line rises from left to right and passes through the points (0, 2.1) and (30, 13.1). All plotted points are on or close to the line. All values are approximate.
C.

A coordinate system has a horizontal axis with tick marks dividing it into 7 regular intervals to the right of the vertical axis and has a vertical axis with tick marks dividing it into 14 intervals above the horizontal axis. The following points are plotted: (0, 2.3); (10, 4.8); (15, 6.5); (20, 7.8); (25, 9.3); (30, 11.1). A line rises from left to right and passes through the points (0, 2.1) and (30, 10.9). All plotted points are on or close to the line. All values are approximate.
D.

A coordinate system has a horizontal axis with tick marks dividing it into 7 regular intervals to the right of the vertical axis and has a vertical axis with tick marks dividing it into 14 intervals above the horizontal axis. The following points are plotted: (0, 2.3); (5, 4.8); (10, 6.5); (15, 7.8); (20, 9.3); (25, 11.1). A line rises from left to right and passes through the points (0, 4.1) and (30, 12.9). All plotted points are on or below the line. All values are approximate.
Part 3
Since the graph of the model is
▼
to the data​ points, the model is​ a(n)
▼
fit to the data.
Part 4
c. At what rate does the model indicate that severe obesity is growing during this​ period?
At
  
enter your response here
percentage point(s) per 5 years
percentage point(s) per year
percentage point(s) per 5 years
year(s) per percentage point
​(Type an integer or decimal rounded to the nearest tenth as​ needed.)
yxx(Type an integer or decimal rounded to three decimal places as needed.)All graphs have viewing window [0,35] by [0,14] with Xscl5 and Yscl1.Since the graph of the model isto the data points, the model is a(n)to the data points, the model is a(n)fit to the data.At(Type an integer or decimal rounded to the nearest tenth as needed.)At units

Answer 1

Part 1: Finding the Linear Function

To find the linear function that models severe obesity, we have the data points for years after 2000 (x) and the corresponding percent of severe obesity (y). We can derive the linear function in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Using the data points:

• (0, 2.3)
• (10, 4.8)
• (15, 6.5)
• (20, 7.8)
• (25, 9.3)
• (30, 11.1)

We can calculate the slope between two points. For instance, between the years 0 (2000) and 10 (2010):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4.8 - 2.3}{10 - 0} = \frac{2.5}{10} = 0.25 \]

However, to find the overall slope that fits all the data points, we can simply use the first and last points:

• (0, 2.3)
• (30, 11.1)

Calculating the slope (m):

\[ m = \frac{11.1 - 2.3}{30 - 0} = \frac{8.8}{30} \approx 0.293 \]

The y-intercept (b) is simply the value of y when x = 0, which is 2.3. Therefore, the linear function is:

\[ y = 0.293x + 2.3 \]

Part 2: Graphing the Model and Comment on the Fit

Given the options for the graph:

Your Choice:

D. "A coordinate system has a horizontal axis with tick marks dividing it into 7 regular intervals to the right of the vertical axis and has a vertical axis with tick marks dividing it into 14 intervals above the horizontal axis. The following points are plotted: (0, 2.3); (5, 4.8); (10, 6.5); (15, 7.8); (20, 9.3); (25, 11.1). A line rises from left to right and passes through the points (0, 4.1) and (30, 12.9). All plotted points are on or below the line. All values are approximate."

Correct Answer:

The answer that correctly indicates the plotted points being on/close to the fitted line is:

C:

• A line rises from left to right and passes through the points (0, 2.1) and (30, 10.9). All plotted points are on or close to the line.

This indicates that there is a good fit between the model and the data points.

Part 3: Comment on the Fit

Since the graph of the model is close to the data points, the model is a(n) good fit to the data.

Part 4: Rate of Growth

The function shows that severe obesity increases by the slope (m), which we calculated as approximately 0.293 percentage points per year.

To find the rate per 5 years, you multiply by 5:

\[ \text{Rate per 5 years} = 0.293 \times 5 \approx 1.465 \]

Final Answer Summary:

• Linear function: \( y = 0.293x + 2.3 \)
• The model is a good fit to the data points.
• The rate of growth of severe obesity is approximately \( 1.5 \) percentage points per 5 years (rounded to one decimal place).