A wire 320 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?
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A wire 320 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?
To solve this problem, let's assign variables to represent the lengths of the wire for each piece.
Let's say the length of the wire used to create the square is x inches, then the length of the wire used to create the circle would be 320 - x inches.
By using these variables, we can now set up equations to represent the relationship between the lengths of the wire and the areas of the corresponding figures.
1. Square:
The length of each side of the square is x/4 (since there are four equal sides in a square).
Therefore, the perimeter of the square (which is equal to the length of wire used) is 4 * (x/4) = x inches.
2. Circle:
The circumference of a circle is equal to the length of wire used. So, the circumference of the circle would be 2 * pi * r, where r is the radius of the circle.
Therefore, the wire length is equal to: 2 * pi * r = 320 - x inches.
Now, let's find the areas of the square and the circle.
3. Area of the square:
The area of a square is equal to the side length squared, so the area of the square would be (x/4)^2 = x^2 / 16 square inches.
4. Area of the circle:
The area of a circle is given by pi * r^2, where r is the radius of the circle.
So, the area of the circle is pi * r^2, and the radius of the circle is given by half the circumference divided by 2pi: (320 - x) / (2 * pi * 2pi).
Thus, the area of the circle is pi * ((320 - x) / (2 * pi * 2pi))^2 = (320 - x)^2 / (8 * pi) square inches.
Since the problem states that the areas of the square and circle are equal, we can equate the two areas and solve for x:
x^2 / 16 = (320 - x)^2 / (8 * pi)
To solve this equation, let's rearrange it:
x^2 / 16 = (320 - x)^2 / (8 * pi)
Multiply both sides by 16:
x^2 = (320 - x)^2 / (pi / 2)
Now, cross multiply:
x^2 * (pi / 2) = (320 - x)^2
Expand the right side:
x^2 * (pi / 2) = 102400 - 640x + x^2
Combine like terms:
x^2 * (pi / 2) - x^2 = 102400 - 640x
Factor out x^2:
x^2 * ((pi / 2) - 1) = 102400 - 640x
Divide both sides by ((pi / 2) - 1):
x^2 = (102400 - 640x) / ((pi / 2) - 1)
Now, take the square root of both sides to solve for x:
x = sqrt((102400 - 640x) / ((pi / 2) - 1))
Using a numerical equation solver or a calculator, we find:
x ≈ 128.1
So, the length of the wire used to create the square is approximately 128.1 inches.
Therefore, the length of the wire used to create the circle is approximately 320 - 128.1 = 191.9 inches.
Hence, to the nearest tenth of an inch, the lengths of the two pieces of wire are 128.1 inches and 191.9 inches, respectively.