A wire 360 cm 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 centimeter)? Take the length of the circle to be x cm.
pi r^2 = s^2
4 s + 2 pi r = 3600
it seems they want you to use x for 2 pi r so
r = x/(2pi)
pi r^2 = x^2/(4 pi) = s^2
so
4 s + x = 360
2x/( sqrt pi) + x = 360
2.13 x = 360
x = 169.1 cm
360 - x = 4s = 190.9 cm
s = 47.7
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check
r = x/(2 pi) = 26.9
pi r^2 = 2276
s^2 = 2277 close enough
To solve this problem, we need to set up equations based on the given information.
Let's start by finding the length of the side of the square.
The perimeter of a square is given by 4 times the length of one side. So, if one piece of wire is formed into a square, and its length is x cm, then the length of each side of the square would be x/4 cm.
Now, let's find the circumference of the circle. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. In this case, the length of the wire used to form the circle is x cm, so the circumference would be equal to x cm.
We know that the two figures have the same area, so we can set up the following equation:
Area of the square = Area of the circle
The area of a square is given by the formula A = s^2, where s is the length of a side. Similarly, the area of a circle is given by the formula A = πr^2.
Setting the two areas equal, we get:
(x/4)^2 = πr^2
Simplifying this equation, we have:
x^2/16 = πr^2
Now, we need to find an equation relating the circumference, x, to the radius, r. The circumference of a circle is also given by the formula C = 2πr. Since the length of the wire used to form the circle is x cm, we can write the equation as:
x = 2πr
Now we have a system of two equations:
x^2/16 = πr^2 (Equation 1)
x = 2πr (Equation 2)
To solve these equations, we can substitute Equation 2 into Equation 1.
Substituting 2πr for x in Equation 1, we have:
(2πr)^2 / 16 = πr^2
Simplifying this equation, we get:
4πr^2 = πr^2
Dividing both sides by πr^2, we have:
4 = 1
This is a contradiction, which means there is no solution to the problem as stated.