In a floor plan, the length, [tex]$l$[/tex], of a rectangular room is twice its width, [tex]$w$[/tex]. The perimeter of the room must be greater than 72 feet.
Which inequality can be used to find all possible widths of the room, in feet?
A. [tex]$6 w\ \textgreater \ 72$[/tex]
B. [tex]$6 w\ \textless \ 72$[/tex]
C. [tex]$3 w\ \textgreater \ 72$[/tex]
D. [tex]$3 w\ \textless \ 72$[/tex]
Answer (1)
Express the length in terms of the width: l = 2 w .
Write the perimeter formula: P = 2 l + 2 w .
Substitute and simplify: P = 6 w .
Form the inequality: 72"> 6 w > 72 , so the answer is 72}"> 6 w > 72 .
Explanation
Understanding the Problem Let's analyze the problem. We are given that the length of a rectangular room is twice its width, which means l = 2 w . We also know that the perimeter of the room must be greater than 72 feet. Our goal is to find an inequality that represents this situation in terms of the width, w .
Substituting the Length The formula for the perimeter of a rectangle is P = 2 l + 2 w . Since l = 2 w , we can substitute this into the perimeter formula to get P = 2 ( 2 w ) + 2 w .
Simplifying the Perimeter Now, let's simplify the expression for the perimeter: P = 4 w + 2 w = 6 w . We know that the perimeter must be greater than 72 feet, so 72"> P > 72 . Substituting P = 6 w into the inequality, we get 72"> 6 w > 72 .
Finding the Inequality Therefore, the inequality that can be used to find all possible widths of the room is 72"> 6 w > 72 . This corresponds to option A.
Examples
Imagine you're designing a rectangular garden where the length must be twice the width. If you want the fence around the garden to be more than 72 feet long, this problem helps you determine the possible widths of the garden. Understanding this helps in planning layouts and managing resources effectively, ensuring your garden meets the desired size and aesthetic requirements. This concept is also applicable in urban planning, interior design, and even in creating layouts for events.
