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 analyse 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

The width to length relationship of the patio described by 2w - 5 is linear, which results in a straight line on a graph. Feet would be the appropriate unit for both width and length, and not all given width values result in sensible patio dimensions.

Step-by-step explanation:

The relationship between the width and the length of the patio is indeed linear, as it can be described by the linear expression 2w − 5, where w represents the width. This kind of relationship, where the length is directly proportional to the width with a constant difference, results in a straight line when plotted on a graph.

When graphing this relationship, it's suitable to use feet as the units for both the x-axis (width) and y-axis (length), with intervals chosen based on the range of dimensions that are practical for a backyard patio. For instance, an interval of 5 feet might be appropriate.

Calculating for the given widths:

• When w = 2 feet, the length is 2(2) − 5 = 4 − 5 = −1 (not practical for a patio dimension).
• When w = 5 feet, the length is 2(5) − 5 = 10 − 5 = 5 feet (a possible patio dimension).
• When w = 10 feet, the length is 2(10) − 5 = 20 − 5 = 15 feet (a possible patio dimension).
• When w = 50 feet, the length is 2(50) − 5 = 100 − 5 = 95 feet (a very large but possible patio dimension).

Not all the calculated lengths make sense in the scenario, specifically where the length calculation results in negative or unrealistic measurements for a patio.