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Question
A length of wire is used to construct two square wire frames.  The area enclosed by one frame is one-half the area enclosed by the other.

Length of wire - 100 inches.

Disregard thickness of wire.

Calculate dimensions of frames.  Thanks.

"A length of wire is used to construct two square wire frames."
Let the smaller square be x by x inches, and the larger square y by y inches.

"The area of one square is one-half the area of the other square."
x² = y²/2

"Length of wire = 100 inches."
(perimeter of smaller square) + (perimeter of larger square) = 100
4x + 4y = 100
x + y = 25
x = 25-y
x² = y² - 50y + 625

y² - 50y + 625 = y²/2
2y² - 100y + 1250 = y²
y² - 100y + 1250 = 0

y = [100 ± √(100² - 4·1·1250)]/(2·1)
= [100 ± √5000]/2
= [100 ± 50√2]/2
= 50 ± 25√2
Discard 50+25√2 because it is longer than the wire.
y = 50 - 25√2 ≅ 14.64 inches
The larger square is approximately 14.64 inches by 14.64 inches.

x = 25-y = -25 + 25√2 ≅ 10.36 inches
The smaller square is approximately 10.36 inches by 10.36 inches.

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