Joe has exactly enough paint to paint the surface of a cube whose main diagonal is 2. It turns out that this is also exactly enough paint to paint the surface of a sphere. Find the radius of the sphere.
+9674 The main diagonal of a cube is \(\sqrt 3\) times longer than its side length.
Therefore, the side length of cube = \(\dfrac2{\sqrt 3}\).
Now, we can calculate the surface area by the formula \(\text{surface area of cube} = 6(\text{side length})^2\).
\(\text{surface area of cube} = 6\left(\dfrac2{\sqrt 3}\right)^2 = 8\)
Since the amount of paint used is exactly the same as the amount of paint to paint the surface of a sphere, the surface area of the sphere is the same as that of the cube.
\(\text{surface area of sphere} = 8\).
Let r be the radius of the sphere. Then,
\(4\pi r^2 = 8\\ r^2 = \dfrac2\pi\\ r = \sqrt{\dfrac2\pi}\)
Therefore, radius of sphere = \(\sqrt{\dfrac{2}\pi}\).
+2668 The main diagonal is 2. Because it is a cube, all the side lengths are the same, and we have the equation: \(\sqrt{x^2+x^2+x^2}=2\)
Solving, we find \(x^2 = {4 \over3}\). The surface area is \(6x^2\), meaning the surace area of the cube is \(8\).
Now, we have the equation: \(4\pi r=8\)
Diving both sides by 4, we get: \(\pi r = 2\)
Solving for r, we find \(r = \color{brown}\boxed{\sqrt{2\over \pi}}\)
