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# 3d geometry

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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.

Apr 16, 2022

#1
+9461
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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 = $$\sqrt{\dfrac{2}\pi}$$.

Apr 16, 2022
#2
+2558
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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 $$r = \color{brown}\boxed{\sqrt{2\over \pi}}$$

Apr 16, 2022