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.
Answers
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.
Step-by-step explanation:
see attachment for answer
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