To determine which pairs of variables have a linear relationship, we need to understand how each variable is related to the others in the context of a cube.
1. **Side length \( s \)**
2. **Perimeter of 1 face**: A cube has square faces. The perimeter of one face (a square) would be \( 4s \).
3. **Area of 1 face**: The area of one face (a square) would be \( s^2 \).
4. **Surface area**: The total surface area of a cube with six faces would be \( 6s^2 \).
5. **Volume**: The volume of a cube would be \( s^3 \).
We'll consider each pair mentioned and identify if they have a linear relationship:
1. **Side length and perimeter of 1 face**:
- Perimeter of 1 face: \( 4s \)
- This is a linear relationship, with the perimeter being directly proportional to the side length.
2. **Perimeter of 1 face and area of 1 face**:
- Perimeter: \( 4s \)
- Area: \( s^2 \)
- To check for linearity, we need to see if the relationship \( P \propto A \) (where \( P \) is perimeter and \( A \) is area) holds. Substituting, we get \( P = 4\sqrt{A} \), which is not a linear relationship.
3. **Surface area and volume**:
- Surface area: \( 6s^2 \)
- Volume: \( s^3 \)
- Note the relationship \( Volume = \left(\frac{\text{Surface area}}{6}\right)^{3/2} \), which is not linear.
4. **Area of 1 face and surface area**:
- Area of 1 face: \( s^2 \)
- Surface area: \( 6s^2 \)
- Surface area is directly proportional to the area of one face with a constant factor of 6, which is a linear relationship.
5. **Side length and volume**:
- Side length: \( s \)
- Volume: \( s^3 \)
- The relationship \( Volume = s^3 \) is not linear.
From our analysis, the pairs of variables that have a linear relationship are:
- **Side length and perimeter of 1 face**
- **Area of 1 face and surface area**
The graphs below show measurements from cubes with different side lengths.
Which pairs of variables have a linear relationship? Select two options.
side length and perimeter of 1 face
perimeter of 1 face and area of 1 face
surface area and volume
area of 1 face and surface area
side length and volume
1 answer