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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

quadratic formula

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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