A wire 360cm is cut into two pieces.One piece is formed into a square and the other is formed into a circle.If two figures have the same area,what are the two lenghts of two pieces of wire( to the nearest hundreth of centimeter?,show all the necessary working clearly.
if the square has side s
and the circle has radius r, then
r+s = 360
pi r^2 = (360-r)^2
(pi-1)r^2 + 720r - 129600 = 0
r = 129.85
...
To solve this problem, we need to first determine the lengths of the wire used for the square and the circle. Let's assign variables to represent these lengths.
Let's call the length of wire used for the square 'x' centimeters.
Therefore, the length of wire used for the circle will be 360 - x centimeters.
The perimeter of a square is equal to 4 times the length of a side. Since our square has four equal sides, each side length would be x/4.
The circumference of a circle is given by the formula 2 * π * r, where r is the radius of the circle. Since wire forms the perimeter of the circle, we can equate it to the circumference, 2 * π * r, and solve for r.
Perimeter of the square = 4 * (x/4) = x
Circumference of the circle = 2 * π * r = 360 - x
We know that the area of the square is equal to the area of the circle. The area of a square is given by the formula side * side, and the area of a circle is given by the formula π * r^2.
Area of the square = (x/4) * (x/4) = x^2/16
Area of the circle = π * r^2 = π * (r^2)
Setting these two equations equal to each other, we can solve for the radius of the circle.
x^2/16 = π * (r^2)
Simplifying further, we get:
(r^2) = (x^2) / (16 * π)
Taking the square root of both sides:
r = √(x^2) / (√(16 * π))
Now we have equations for both the perimeter of the square and the circumference of the circle:
Perimeter of the square = x
Circumference of the circle = 2 * π * r
Since the area of the square is equal to the area of the circle, we can equate the areas:
Area of the square = Area of the circle
(x/4)^2 = π * r^2
Substituting the value of r, we get:
(x/4)^2 = π * (√(x^2) / (√(16 * π)))^2
Simplifying further:
(x/4)^2 = π * (x^2) / (16 * π)
(x^2) / 16 = (x^2) / 16
Now, we can solve for x. Multiply through by 16 to remove the fractions:
x^2 = x^2
This equation implies that x can be any value. Therefore, the lengths of the two pieces of wire can vary as long as their total is 360cm.