Question

donna wants to paint a rectangular region of a wall with one layer of paint. she has 20 ft of tape to tape around the border of the region she paints. she is considering three regions with widths of 3 ft, 5 ft, and 6 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.

	width	length	area
region #2	5 ft	\\(\square\\) ft	\\(\square\\) \\(\text{ft}^2\\)
region #3	6 ft	\\(\square\\) ft	\\(\square\\) \\(\text{ft}^2\\)

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

Explanation:

Step1: Recall perimeter formula for rectangle

The perimeter of a rectangle is $P = 2(l + w)$, where $P$ is perimeter, $l$ is length, $w$ is width. Given $P=20$ ft, rearrange to find $l$: $l = \frac{P}{2} - w$

Step2: Calculate length for Region #1

Substitute $w=3$ ft:
$l = \frac{20}{2} - 3 = 10 - 3 = 7$ ft

Step3: Calculate area for Region #1

Area formula: $A = l \times w$
$A = 7 \times 3 = 21$ ft²

Step4: Calculate length for Region #2

Substitute $w=5$ ft:
$l = \frac{20}{2} - 5 = 10 - 5 = 5$ ft

Step5: Calculate area for Region #2

$A = 5 \times 5 = 25$ ft²

Step6: Calculate length for Region #3

Substitute $w=6$ ft:
$l = \frac{20}{2} - 6 = 10 - 6 = 4$ ft

Step7: Calculate area for Region #3

$A = 4 \times 6 = 24$ ft²

Step8: Compare areas to find maximum

Compare 21, 25, 24. The largest is 25 ft² (Region #2)

Answer:

(a) Completed Table:

	Width	Length	Area
Region #2	5 ft	5 ft	25 ft²
Region #3	6 ft	4 ft	24 ft²

(b)

Region #2