The Phillips family wants to fence their backyard. They know the yard has a perimeter of 24 meters and an area of 32 square meters. What is the yard's length and width?
Assuming that the yard is a rectangle
Area = length × width
32 = length × width
Perimeter = 2 × length + 2 × width
24 = 2 × length + 2 × width
Using 32 = length × width, width = 32 / length
Replace width = 32 / length in 24 = 2 × length + 2 × width
We get:
24 = 2 × length + 2 × (32 / length)
24 = 2length + 64 / length
24 - 2length = 64 / length
Multiply both sides by length
24 × length - 2length × length = ( 64 × length ) / length
24length - 2length^{2} = 64
24length - 24length - 2length^{2} = 64 - 24length
0 - 2length^{2} = + 64 - 24length
0 - 2length^{2} + 2length^{2} = 64 - 24length + 2length^{2}
0 + 0 + = 64 - 24length + 2length^{2}
2length^{2} - 24length + 64 = 0
Divide both sides by 2
length^{2} - 12length + 32 = 0
Look for factors of 32 that will add up to -12
-8 × -4 = 32 and -8 + -4 = -12
So our factors are -8 and -4
We get (length + -8) × (length + -4) = 0
length + -8 = 0
length + -4 = 0
length = 8
length = 4
If length = 8, width = 32 / length = 32 / 8 = 4
If length = 4, width = 32 / length = 32 / 4 = 8
Thus, either the length = 4 and the width = 8 or the length = 8 and the width = 4
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