Question
The location of runway A for a new airport is shown on the coordinate plane.
Two additional runways are planned. Each will pass through the point (0, -4) on the coordinate plane.
Runway B is to be parallel to runway A.
Runway C is to be perpendicular to runway A.
Which equations could describe either of those runways? Choose all that are correct.
(2 points)
Responses
y=3/4x−4
y=−4/3x−4
y=4/3x−4
y=−3/4x−4
Use the data in the table to create a scatter plot, and then find the correlation coefficient.
x y
3 7
3 9
4
6
6 11
7 10
8 15
8 16
9 14
10 12
(1 point)
Responses
r=0.79
r=4.27
r=0.63
r=1.06
Two additional runways are planned. Each will pass through the point (0, -4) on the coordinate plane.
Runway B is to be parallel to runway A.
Runway C is to be perpendicular to runway A.
Which equations could describe either of those runways? Choose all that are correct.
(2 points)
Responses
y=3/4x−4
y=−4/3x−4
y=4/3x−4
y=−3/4x−4
Use the data in the table to create a scatter plot, and then find the correlation coefficient.
x y
3 7
3 9
4
6
6 11
7 10
8 15
8 16
9 14
10 12
(1 point)
Responses
r=0.79
r=4.27
r=0.63
r=1.06
Answers
GPT-4o mini
To determine the equations that describe runways B and C based on runway A, we first need to establish the characteristics of these runways regarding their slopes.
### Runway B (parallel to Runway A)
Parallel lines have the same slope. Therefore, any equation for Runway B must have the same slope as Runway A. Without knowing the specific slope of runway A, I cannot definitively answer regarding which choices are valid for Runway B.
### Runway C (perpendicular to Runway A)
Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, if the slope of runway A is \( m \), then the slope of runway C would be \( -\frac{1}{m} \).
Let's evaluate the given equations:
1. **y = \( \frac{3}{4}x - 4 \)** (slope = \( \frac{3}{4} \))
- Negative reciprocal: \( -\frac{4}{3} \) (could describe runway C)
2. **y = \( -\frac{4}{3}x - 4 \)** (slope = \( -\frac{4}{3} \))
- Negative reciprocal: \( \frac{3}{4} \) (could describe runway B)
3. **y = \( \frac{4}{3}x - 4 \)** (slope = \( \frac{4}{3} \))
- Negative reciprocal: \( -\frac{3}{4} \) (could describe runway C)
4. **y = \( -\frac{3}{4}x - 4 \)** (slope = \( -\frac{3}{4} \))
- Negative reciprocal: \( \frac{4}{3} \) (could describe runway B)
Assuming the slopes of runway A might be either \( \frac{3}{4} \) or \( -\frac{3}{4} \), the equations that could describe either runway B or C would be:
- **y = \( -\frac{4}{3}x - 4 \)** (Runway B)
- **y = \( \frac{4}{3}x - 4 \)** (Runway C)
- **y = \( -\frac{3}{4}x - 4 \)** (Runway B)
- **y = \( \frac{3}{4}x - 4 \)** (Runway C)
### Scatter Plot Correlation Coefficient
To find the correlation coefficient \( r \), we first need to calculate it using the data provided. You can do this by performing the following steps or use a calculator/statistical software:
1. Calculate the means of x and y.
2. Find the products of deviations from the mean for both x and y.
3. Divide the result by the product of the standard deviations of x and y.
However, based on your multiple-choice options, we should recognize reasonable estimates of correlation based on the provided data points. Without direct calculations of each data point’s corresponding correlation coefficient, we can analyze the closeness of the points visually.
If we follow through with standard deviation and product calculations, the final correlation coefficient value \( r \) will fall within -1 to 1, and among your provided options, **r = 0.79** seems to indicate a strong positive correlation based on the data provided, where variables appear closely clustered and increasing as x increases.
Ultimately, the answers are:
- For the runway equations: **y = \( -\frac{4}{3}x - 4 \)** and **y = \( \frac{4}{3}x - 4 \)** or **y = \( -\frac{3}{4}x - 4 \)** and **y = \( \frac{3}{4}x - 4 \)**
- For the correlation coefficient: **r = 0.79**
### Runway B (parallel to Runway A)
Parallel lines have the same slope. Therefore, any equation for Runway B must have the same slope as Runway A. Without knowing the specific slope of runway A, I cannot definitively answer regarding which choices are valid for Runway B.
### Runway C (perpendicular to Runway A)
Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, if the slope of runway A is \( m \), then the slope of runway C would be \( -\frac{1}{m} \).
Let's evaluate the given equations:
1. **y = \( \frac{3}{4}x - 4 \)** (slope = \( \frac{3}{4} \))
- Negative reciprocal: \( -\frac{4}{3} \) (could describe runway C)
2. **y = \( -\frac{4}{3}x - 4 \)** (slope = \( -\frac{4}{3} \))
- Negative reciprocal: \( \frac{3}{4} \) (could describe runway B)
3. **y = \( \frac{4}{3}x - 4 \)** (slope = \( \frac{4}{3} \))
- Negative reciprocal: \( -\frac{3}{4} \) (could describe runway C)
4. **y = \( -\frac{3}{4}x - 4 \)** (slope = \( -\frac{3}{4} \))
- Negative reciprocal: \( \frac{4}{3} \) (could describe runway B)
Assuming the slopes of runway A might be either \( \frac{3}{4} \) or \( -\frac{3}{4} \), the equations that could describe either runway B or C would be:
- **y = \( -\frac{4}{3}x - 4 \)** (Runway B)
- **y = \( \frac{4}{3}x - 4 \)** (Runway C)
- **y = \( -\frac{3}{4}x - 4 \)** (Runway B)
- **y = \( \frac{3}{4}x - 4 \)** (Runway C)
### Scatter Plot Correlation Coefficient
To find the correlation coefficient \( r \), we first need to calculate it using the data provided. You can do this by performing the following steps or use a calculator/statistical software:
1. Calculate the means of x and y.
2. Find the products of deviations from the mean for both x and y.
3. Divide the result by the product of the standard deviations of x and y.
However, based on your multiple-choice options, we should recognize reasonable estimates of correlation based on the provided data points. Without direct calculations of each data point’s corresponding correlation coefficient, we can analyze the closeness of the points visually.
If we follow through with standard deviation and product calculations, the final correlation coefficient value \( r \) will fall within -1 to 1, and among your provided options, **r = 0.79** seems to indicate a strong positive correlation based on the data provided, where variables appear closely clustered and increasing as x increases.
Ultimately, the answers are:
- For the runway equations: **y = \( -\frac{4}{3}x - 4 \)** and **y = \( \frac{4}{3}x - 4 \)** or **y = \( -\frac{3}{4}x - 4 \)** and **y = \( \frac{3}{4}x - 4 \)**
- For the correlation coefficient: **r = 0.79**