+45 1. Three squares are drawn inside a circle as shown. The area of the circle is 60pi square inches. How many square inches are in the area of one square?
2. Two circles inside a square are externally tangent to each other and are tangent to certain sides of the square as shown. The perimeter of the square is \(2 \sqrt{2}\) . What is the sum of the circumferences of the two circles?
+37153 Here is A way to solve first one....may not be THE BEST way....
circle radius = sqrt 60
draw a line from origin to corne of square that touches circle this line forms a arc tan (1/3) = 18.435 degree angle
sqrt 60 cos 18.435 = length of 3 boxes = 7.34 so each square is side length 7.34/3
then total area of 1 square = (7.34/3 * 7.34/3) = 6 in2
+506 ElectricPavlov nice solution!
Solution for 2:
Since the perimeter of the square is \(2\sqrt{2}\), the side length of one side of the square is \(\frac{\sqrt{2}}{2}\). The diagonal of that square is then equal to \(\frac{\sqrt{2}}{2} \cdot \sqrt{2} = 1\)
Let x be the radius of one circle, and let y be the radius of the other circle.
Notice that the diagonal is equal to \(x\sqrt{2} + x + y\sqrt{2} + y\). Therefore,
\(x\sqrt{2} + x + y\sqrt{2} + y = 1\\ x(\sqrt{2}+1)+ y(\sqrt{2}+1)=1\\ x+y=\frac{1}{\sqrt{2}+1} = \sqrt{2}-1\)
Since the sum of the circumference of the circle is 2 times the radius times pi, just multiply 2pi to get the final answer:
\(\boxed{2\sqrt{2}\pi-2\pi}\)
+506 I'm really sorry that I got it wrong. Could you tell me the answer so I can see where I made a mistake?
Edit: It seems like you mistyped the problem according to this:
https://web2.0calc.com/questions/help_12091
The solution would have been:
\(x(\sqrt{2}+1)+y(\sqrt{2}+1)=\frac{1}{2}(\sqrt{2}+1)\\ x+y=\frac{1}{2}\)
Since the sum of the circumference of the circle is 2 times the radius times pi, just multiply 2pi to get the final answer:
\(\frac{1}{2} \cdot 2 \cdot \pi = \boxed{\pi}\)
