Question

david wants to paint a rectangular region of a wall with one layer of paint. he has 12 ft of tape to tape around the border of the region he paints. he is considering three regions with widths of 3 ft, 4 ft, and 5 ft. answer the questions below to find which of these regions would require the most paint. (a) fill in the table to find the length and the area for each region.

region #1	region #2	region #3
length	ft	ft	ft
area	ft²	ft²	ft²

(b) which of these regions would require the most paint?
○ region #1
○ region #2
○ region #3

Explanation:

Step1: Recall rectangle - area formula

The area formula for a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. We know the perimeter (the amount of tape) is related to the sum of the sides of the rectangle. Since we have 12 ft of tape and for a rectangle with one - side against a wall, if the width is $w$ and the length is $l$, the perimeter formula is $P=l + 2w$ (assuming the non - wall side is the length). Given $P = 12$, so $l=12 - 2w$.

Step2: Calculate area for each region

Region #1:

$w = 3$ ft. Then $l=12-2\times3=6$ ft. The area $A_1=l\times w=6\times3 = 18$ ft².

Region #2:

$w = 4$ ft. Then $l=12-2\times4 = 4$ ft. The area $A_2=l\times w=4\times4=16$ ft².

Region #3:

$w = 5$ ft. Then $l=12-2\times5=2$ ft. The area $A_3=l\times w=2\times5 = 10$ ft².

Answer:

Region	Width	Length	Area
Region #2	4 ft	4 ft	16 ft²
Region #3	5 ft	2 ft	10 ft²

(b) Region #1