Buddy is going to decorate his rectangular backyard with holiday lights. The area of his backyard is 72 square feet and the perimeter is 34 feet. If he wants to use one color for the widths and another color for the lengths, how long does each color on each side need to be to cover the entire perimeter of his backyard?

The length of the color of each side has to be 9 ft long and 8 ft wide or 8 ft long and 9 ft wide for it to be able to cover the entire perimeter of the wall.

Step-by-step explanation:

Let l be the length of the wall and b its breadth. Its area A = lb = 72 ft² (1)

Also, its perimeter P = 2(l + b) = 34 ft

l + b = 17 (2)

From equation (1),

l = 72/b

substituting this into (2), we have

72/b + b = 17

multiplying through by b, we have

72 + b² = 17b

simplifying, we have

b² - 17b + 72 = 0

We find b by factorizing

So, b² - 8b - 9b + 72 = 0

b(b - 8) -9(b - 8) = 0

(b - 8)(b - 9) = 0

b - 8 = 0 or b - 9 =0

b = 8 or b = 9

Substituting b = 8 into l, we have

l = 72/b = 72/8 = 9

Substituting b = 9 into l, we have

l = 72/b = 72/9 = 8

So, the length of the color of each side has to be 9 ft long and 8 ft wide or 8 ft long and 9 ft wide for it to be able to cover the entire perimeter of the wall.