Answer :
Let's delve into this problem step-by-step to determine the slopes of the lines for Restaurant A and Restaurant B, and then compare them.
### Calculate the Slope for Restaurant A
For Restaurant A, we have the points [tex]\((10, 1)\)[/tex], [tex]\((20, 2)\)[/tex], and [tex]\((30, 3)\)[/tex].
To find the slope of the line (denoted as [tex]\(m_a\)[/tex]) passing through these points, we can use the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the first two points [tex]\((10, 1)\)[/tex] and [tex]\((20, 2)\)[/tex]:
[tex]\[ m_a = \frac{2 - 1}{20 - 10} = \frac{1}{10} = 0.1 \][/tex]
### Calculate the Slope for Restaurant B
For Restaurant B, we have the points [tex]\((25, 5)\)[/tex], [tex]\((50, 10)\)[/tex], and [tex]\((75, 15)\)[/tex].
Using the first two points [tex]\((25, 5)\)[/tex] and [tex]\((50, 10)\)[/tex]:
[tex]\[ m_b = \frac{10 - 5}{50 - 25} = \frac{5}{25} = \frac{1}{5} = 0.2 \][/tex]
### Compare the Slopes
We now have the slopes:
- Slope for Restaurant A: [tex]\(m_a = 0.1\)[/tex]
- Slope for Restaurant B: [tex]\(m_b = 0.2\)[/tex]
To find out how many times greater the slope of Restaurant B ([tex]\(m_b\)[/tex]) is compared to the slope of Restaurant A ([tex]\(m_a\)[/tex]), we can divide [tex]\(m_b\)[/tex] by [tex]\(m_a\)[/tex]:
[tex]\[ \text{Comparison} = \frac{m_b}{m_a} = \frac{0.2}{0.1} = 2.0 \][/tex]
### Conclusion
The slope of the line for Restaurant B is 2 times greater than the slope of the line for Restaurant A.
Hence, the correct comparison is: The slope of the line for Restaurant B is 2 times greater than the slope of the line for Restaurant A.
### Calculate the Slope for Restaurant A
For Restaurant A, we have the points [tex]\((10, 1)\)[/tex], [tex]\((20, 2)\)[/tex], and [tex]\((30, 3)\)[/tex].
To find the slope of the line (denoted as [tex]\(m_a\)[/tex]) passing through these points, we can use the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the first two points [tex]\((10, 1)\)[/tex] and [tex]\((20, 2)\)[/tex]:
[tex]\[ m_a = \frac{2 - 1}{20 - 10} = \frac{1}{10} = 0.1 \][/tex]
### Calculate the Slope for Restaurant B
For Restaurant B, we have the points [tex]\((25, 5)\)[/tex], [tex]\((50, 10)\)[/tex], and [tex]\((75, 15)\)[/tex].
Using the first two points [tex]\((25, 5)\)[/tex] and [tex]\((50, 10)\)[/tex]:
[tex]\[ m_b = \frac{10 - 5}{50 - 25} = \frac{5}{25} = \frac{1}{5} = 0.2 \][/tex]
### Compare the Slopes
We now have the slopes:
- Slope for Restaurant A: [tex]\(m_a = 0.1\)[/tex]
- Slope for Restaurant B: [tex]\(m_b = 0.2\)[/tex]
To find out how many times greater the slope of Restaurant B ([tex]\(m_b\)[/tex]) is compared to the slope of Restaurant A ([tex]\(m_a\)[/tex]), we can divide [tex]\(m_b\)[/tex] by [tex]\(m_a\)[/tex]:
[tex]\[ \text{Comparison} = \frac{m_b}{m_a} = \frac{0.2}{0.1} = 2.0 \][/tex]
### Conclusion
The slope of the line for Restaurant B is 2 times greater than the slope of the line for Restaurant A.
Hence, the correct comparison is: The slope of the line for Restaurant B is 2 times greater than the slope of the line for Restaurant A.
