Exercise 59 Page 161

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 $4.$ 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. 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. 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.