Question

The Godari family is planning to build a new patio in their backyard. They haven’t decided how big it will be, but they know they don’t want a square patio. One of the family members, Arti, proposes that they define the relationship between the length and the width of the patio this way: the length of the patio will be 5 feet less than twice the width. That means she can represent the length by the expression 2w − 5, where w is the width of the patio. The potential dimensions of the patio are given in whole feet.
1. Is the relationship between the width and the length of the patio linear? Why or why not?

2. When graphing this relationship, what would be appropriate units and intervals to use along the x- and y-axes?

3. Some values of width are shown in the table. Find the corresponding length. Then analyze these values and explain why some of them would make sense in the scenario and others would not.
Width (w) 2 5 10 50
Length (2w − 5)
Analysis

Answer 1

Let the letter $l$ represent the length of the patio. From the question we already know that the relationship between the length and the width ($w$) of the patio are:

$l=2w-5$

1. The relationship between the width and the length is linear because it involves only the power/exponent of 1. In other words, $l$ is raised to the power 1 and $w$ too is raised to the power of 1 in the equation $l=2w-5$.

2. Because both the width and the length are in feet, when graphing this relationship, the appropriate units would be feet.

We may take 1 cm to be 1 ft in both the x and the y axes.

3. For this part a table has been attached.

As can be seen from the table, width of 2 ft gives us length to be -1 ft and that is an absurdity.

Width of 10 ft gives a length of 15 ft which does make sense and that is acceptable because we do not want a square patio and means width and length should not be the same. Same is the analysis for width of 50 feet. We get the length to be 95 feet. Again, the patio will not be a square but a rectangle as is the requirement.

Thus, for widths of 10 and 50 only is the Godari Family's patio possible from the given data and the given relationship between the length and the width.