A linear relationship means that we can represent the relationship between two variables with a straight line.
Perimeter of a Square:
Perimeter of a Square
Side length
Perimeter
1
4
2
8
3
12
4
16
Linearity: Linear. Explanation: See solution.
Area of a Square:
Area of a Square
Side length
Area
1
1
2
4
3
9
4
16
Linearity: Non-linear. Explanation: See solution.
Volume of a Cube:
Volume of a Cube
Side length
Volume
1
1
2
8
3
27
4
64
Linearity: Non-linear. Explanation: See solution.
Practice makes perfect
We have been asked to fill in the given tables and then see whether or not any of them show a linear relationship. A linear relationship means that we can represent the relationship between two variables with a straight line. Let's get started!
Perimeter of a Square
The perimeter of a square P can be found by multiplying its side length by 4.
P=4a
In this formula, a represents the side length of the square and 4 represents the number of sides of the square. Let's fill in the table according to this information.
Perimeter of a Square
Side length
Calculation
Perimeter
1
4( 1)
4
2
4( 2)
8
3
4( 3)
12
4
4( 4)
16
How do we determine whether the table shows a linear relationship? Let's plot the values where x represents the side length and y represents the perimeter and connect the points.
As we can see, we are able to represent the relationship between the side length and perimeter with a line. Therefore, the table shows a linear relationship.
Area of a Square
In this part, we will evaluate the area of a square with given side lengths. Recall that we calculate the area of a square by squaring its side length.
A=s^2
In this formula, A represents the area and s represents the side length. Now, let's fill in the table.
Area of a Square
Side length
Calculation
Area
1
1^2
1
2
2^2
4
3
3^2
9
4
4^2
16
Next, let's try to represent these values on the coordinate plane and connect them.
Since we cannot represent the relationship between the area of a square and its side lengths with a straight line, the table shows a non-linear relationship.
Volume of a Cube
In the last table, we will calculate the volume of a cube by cubing its side length.
V=s^3
In this equation, V is the volume and s is the side length of the cube. Let's fill in the table.
Volume of a Cube
Side length
Calculation
Volume
1
1^3
1
2
2^3
8
3
3^3
27
4
4^3
64
Now, we will check whether these values lie on a straight line.
The values do not lie on a straight line. Thus, the table shows a non-linear relationship.
