Question
Melinda works at a cafe. Each day that she works, she records x, the total dollar amount of her customers’ bills and then y, her total daily wages. The table shows her data for 2 weeks.
A 2-column table with 11 rows. The first column is labeled x with entries 50, 100, 75, 80, 90, 140, 150, 95, 125, 160, 165. The second column is labeled y with entries 36, 43, 38, 40, 42, 50, 60, 43, 46, 50, 55. A graph shows the horizontal axis numbered 40 to 160 and the vertical axis numbered 10 to 60. A line increases from 0 to 160.
According to the line of best fit, what is the minimum amount, to the nearest dollar, Melinda will earn for each day of work, even if she serves no customers?
$18
$26
$36
$40
A 2-column table with 11 rows. The first column is labeled x with entries 50, 100, 75, 80, 90, 140, 150, 95, 125, 160, 165. The second column is labeled y with entries 36, 43, 38, 40, 42, 50, 60, 43, 46, 50, 55. A graph shows the horizontal axis numbered 40 to 160 and the vertical axis numbered 10 to 60. A line increases from 0 to 160.
According to the line of best fit, what is the minimum amount, to the nearest dollar, Melinda will earn for each day of work, even if she serves no customers?
$18
$26
$36
$40
Answers
GPT-4o mini
To determine Melinda's minimum earnings (the y-intercept) when she serves no customers (x = 0), we need to analyze the relationship between x (the total dollar amount of her customers' bills) and y (her total daily wages) using the line of best fit.
To find the y-intercept (the value of y when x = 0), we can apply the concept of slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
where:
- \( m \) is the slope of the line
- \( b \) is the y-intercept
To calculate the slope \( m \), we can use two points from the provided data. For example, let's use the points (50, 36) and (165, 55).
1. **Calculate the slope (m)**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{55 - 36}{165 - 50} = \frac{19}{115} \approx 0.1652
\]
2. **Use one of the points to find the y-intercept (b)**:
We can use the point (50, 36) for this calculation:
\[
36 = 0.1652(50) + b
\]
\[
36 = 8.26 + b
\]
\[
b = 36 - 8.26 = 27.74
\]
To round this to the nearest dollar, we find that the y-intercept \( b \) is approximately \( 28 \).
However, most likely Melinda's minimum earnings for each day (when x = 0) would be a bit higher than $18 and lower than $36 based on the options given. The best estimate based on rounding indicates that:
- The potential minimum earnings to the nearest dollar could be \( 26 \).
Therefore, the answer is:
\[ \boxed{26} \]
To find the y-intercept (the value of y when x = 0), we can apply the concept of slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
where:
- \( m \) is the slope of the line
- \( b \) is the y-intercept
To calculate the slope \( m \), we can use two points from the provided data. For example, let's use the points (50, 36) and (165, 55).
1. **Calculate the slope (m)**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{55 - 36}{165 - 50} = \frac{19}{115} \approx 0.1652
\]
2. **Use one of the points to find the y-intercept (b)**:
We can use the point (50, 36) for this calculation:
\[
36 = 0.1652(50) + b
\]
\[
36 = 8.26 + b
\]
\[
b = 36 - 8.26 = 27.74
\]
To round this to the nearest dollar, we find that the y-intercept \( b \) is approximately \( 28 \).
However, most likely Melinda's minimum earnings for each day (when x = 0) would be a bit higher than $18 and lower than $36 based on the options given. The best estimate based on rounding indicates that:
- The potential minimum earnings to the nearest dollar could be \( 26 \).
Therefore, the answer is:
\[ \boxed{26} \]