a wire of 16 inches cut into 2 pieces.
Each piece bent into a square.
Find the length of the two pieces so the sum of the areas of the two squares in 10 square inches.
L = Length of a wire
x = Length of the first pice
y = Length of the cecond pice
x / 4 = Length of the first square
y / 4 = Length of the second square
L = 16 in
16 = x + y
y = 16 - x
Area of the first pice = ( x / 4 ) ^ 2
Area of the second pice = ( y / 4 ) ^ 2
( x / 4 ) ^ 2 + ( y / 4 ) ^ 2 = 10
x ^ 2 / 16 + y ^ 2 / 16 = 10
( x ^ 2 + y ^ 2 ) / 16 = 10 Multiply both sides with 16
x ^ 2 + y ^ 2 = 160
x ^ 2 + ( 16 - x ) ^ 2 = 160
( Remark: ( 16 - x ) ^ 2 = 16 ^ 2 - 32 x + x ^ 2
x ^ 2 + 16 ^ 2 - 32 x + x ^ 2 = 160
2 x ^ 2 + 256 - 32 x = 160
2 x ^ 2 + 256 - 160 - 32 x = 0
2 x ^ 2 + 96 - 32 x = 0
2 x ^ 2 - 32 x + 96 = 0 Divide both sides with 2
x ^ 2 - 16 x + 48 = 0
The exact solutions are:
x = 12
and
x = 4
Length of a wire = 16 in
y = 16 - x
When: x = 12 ; y = 16 - 12 = 4
When x = 4 ; y = 16 - 4 = 12
Length of the two pieces :
12 in
and
4 in
Proof:
( 12 / 4 ) ^ 2 + ( 4 / 4 ) ^ 2 = 10
3 ^ 2 + 1 ^ 2 = 10
9 + 1 = 10 in ^ 2
P.S.
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x ^ 2 - 16 x + 48 = 0
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x ^ 2 - 16 x + 48 = 0
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To solve this problem, we need to break it down into smaller steps. Let's start by defining the variables:
Let x be the length of the first piece (in inches).
Let y be the length of the second piece (in inches).
We are given that a wire of 16 inches is cut into two pieces, so we can write the equation:
x + y = 16 (Equation 1)
The first piece is bent into a square, so the side length of the first square is x/4.
The second piece is also bent into a square, so the side length of the second square is y/4.
We know that the sum of the areas of the two squares is equal to 10 square inches. So we can write the equation for the area:
(x/4)^2 + (y/4)^2 = 10 (Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of x and y.
Let's solve Equation 1 for x:
x = 16 - y
Substituting this into Equation 2:
((16 - y)/4)^2 + (y/4)^2 = 10
Simplifying:
(16 - y)^2/16 + y^2/16 = 10
Multiplying both sides by 16 (to eliminate the denominators):
(16 - y)^2 + y^2 = 160
Expanding:
256 - 32y + y^2 + y^2 = 160
Rearranging terms:
2y^2 - 32y + 96 = 0
Dividing both sides by 2:
y^2 - 16y + 48 = 0
This is a quadratic equation that can be factored as:
(y - 4)(y - 12) = 0
Setting each factor equal to zero:
y - 4 = 0 or y - 12 = 0
Solving for y:
y = 4 or y = 12
If y is equal to 4, then substituting back into Equation 1 gives us:
x + 4 = 16
x = 12
If y is equal to 12, then:
x + 12 = 16
x = 4
So, there are two possible solutions:
1) x = 12 inches, y = 4 inches
2) x = 4 inches, y = 12 inches
Therefore, the length of the two pieces could be 12 inches and 4 inches, or 4 inches and 12 inches.