To analyze this problem, let's break down the steps needed to identify the correct statement.
1. Understanding the Data:
- The table provides pairs of values (minutes of exercise, calories burned).
- We have points: (15, 75), (30, 150), (45, 225), (60, 300), and (75, 375).
2. Determining the Rate of Change (Slope):
- Using two points from the table, we can calculate the rate of change.
- Let's use the points (15, 75) and (30, 150).
- The formula for the rate of change (slope) is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Substituting the values:
[tex]\[
\text{slope} = \frac{150 - 75}{30 - 15} = \frac{75}{15} = 5
\][/tex]
- Therefore, the rate of change is 5 calories burned per minute.
3. Identifying the \( y \)-intercept:
- The \( y \)-intercept is the point where the line crosses the \( y \)-axis (when \( x = 0 \)).
- Given the pattern of the data points, if there were 0 minutes of exercise, 0 calories would be burned. Thus, the \( y \)-intercept is (0, 0).
4. Formulating the Linear Function:
- With a known slope of 5 and the \( y \)-intercept at (0, 0), the linear function can be formulated as follows:
[tex]\[
y = 5x
\][/tex]
- This represents the relationship between the minutes of exercise (\( x \)) and the calories burned (\( y \)).
5. Reviewing the Statements:
- The \( y \)-intercept of the function is [tex]$(15,75)$[/tex]. This is false; the \( y \)-intercept, where \( x = 0 \), is [tex]$(0,0)$[/tex].
- The rate of change is 15 calories burned per minute. This is false; the rate of change is 5 calories burned per minute.
- A linear function to model the calories burned, \( y \), as a function of time in minutes, \( x \), is \( y = 5x \). This is true.
- An exponential function to model the calories burned, \( y \), as a function of time in minutes, \( x \), is \( y = 5x \). This is false; the function provided is linear, not exponential.
Conclusion:
The correct statement is: A linear function to model the calories burned, [tex]\( y \)[/tex], as a function of time in minutes, [tex]\( x \)[/tex], is [tex]\( y = 5x \)[/tex].
