Question

A piece of wire is 8m.in length is cut into two pieces ,and each piece is bent into a square .where should the cut in the wire be made if the sum of the areas of these squares is to be 2m.

Answer 1

Answer :-

Cut in the wire should be be made exactly middle of the wire.

Explanation :-

Let the length of one of the piece of wire be 'x' m

Let the length of the another piece of wire be 'y' m

Length of the wire = 8 m

i.e Length of one of the piece of wire + Length of another piece of wire = 8 m

⇒ x + y = 8

⇒ y = 8 - x

i.e Length of another piece of wire = (8 - x) m

Each wire is bent into a square

Here

Perimeter of one square = x m

⇒ 4s1 = x m

⇒ s1 = x/4 m

i.e Side of one square (s1) = (x/4) m

Perimeter of another square = (8 - x) m

⇒ 4s2 = (8 - x) m

⇒ s2 = (8 - x)/4 m

i.e Side of another square (s2) = (8 - x)/4 m

Given

Sum of the areas of the squares = 2 m²

Area of one square + Area of another square = 2 m²

⇒ (s1)² + (s2)² = 2 m²

⇒ (x/4)² + {(8 - x)/4}² = 2

⇒ (x²/4²) + {(8 - x)²/4²} = 2

⇒ (x²/16) + {(8² - 2(8)(x) + x²)/16} = 2

⇒ (x²/16) + {(64 - 16x + x²)/16} = 2

⇒ {(x² + 64 - 16x + x²)/16} = 2

⇒ 2x² - 16x + 64 = 2(16)

⇒ 2x² - 16x + 64 = 32

⇒ 2x² - 16x + 64 = 32

⇒ 2x² - 16x + 64 - 32 = 0

⇒ 2x² - 16x + 32 = 0

⇒ 2(x² - 8x + 16) = 0

⇒ x² - 8x + 16 = 0/2

⇒ x² - 8x + 16 = 0

Splitting the middle term

⇒ x² - 4x - 4x + 16 = 0

⇒ x(x - 4) - 4(x - 4) = 0

⇒ (x - 4)(x - 4) = 0

⇒ (x - 4)² = 0

⇒ x - 4 = √0

⇒ x - 4 = 0

⇒ x = 4

Length of the one of the piece of the wire = x = 4 m

Length of another piece of the wire = (8 - x) = (8 - 4) = 4 m

Therefore the cut in the wire should be be made exactly middle of the wire.

Answer 2

Step-by-step explanation:

see attachment for answer

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