Answer:
The piece of wire cut for the square is 180.3 inches, and the piece of wire cut for the circle is 159.7 inches.
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Explanation:
Use the following formulas for the areas of a square and circle:
Square
[tex]A = s^{2}[/tex]
Circle
[tex]A = \pi r^{2}[/tex]
s represents the length of the side in a square, and r represents the radius of the circle.
Set these two equations to be equal to each other.
[tex]s^{2} = \pi r^{2}[/tex]
We can simplify this by square rooting both sides to get s by itself:
[tex]s = \sqrt{\pi r^{2}} [/tex]
Now we'll find the perimeter of the square and the circumference of the circle. Use the following formulas for the perimeter and circumference:
Square
[tex]P = 4s[/tex]
Circle
[tex]C = 2 \pi r[/tex]
The perimeter and circumference must both equal 340, so set the equations to add together to become 340:
[tex]4s + 2 \pi r = 340[/tex]
Because we know what s equals, we can plug it into this equation:
[tex]4( \sqrt{\pi r^{2}} ) + 2 \pi r = 340[/tex]
Simplify the equation by dividing both sides by 2 and extracting r from the first term.
[tex]2r( \sqrt{\pi} ) + \pi r = 170[/tex]
Assuming pi = 3.1416, replace pi with this decimal value and solve:
[tex] \sqrt{3.1416} = 1.772[/tex]
[tex]2r(1.772) + 3.1416r = 170[/tex]
[tex]3.544r + 3.1416r = 170[/tex]
[tex]6.6856r = 170[/tex]
[tex]170 \div 6.6856 = 25.4277[/tex]
[tex]r = 25.4277[/tex]
Rounded to the nearest tenth, the radius of the circle is 25.4.
We can now use the radius to find the circumference of the circle:
[tex]2\pi(25.42)[/tex]
[tex]50.84(3.1416) = 159.7189[/tex]
Rounded to the nearest tenth, the circumference of the circle is 159.7 inches.
Plug this value into the original equation for the perimeter and circumference:
[tex]4s + 159.7 = 340[/tex]
Subtract 159.7 from both sides.
[tex]4s = 180.3[/tex]
The perimeter of the square is 180.3 inches.
