# Is this related to Kepler?

Geometry Level 4

In a cuboid made of glass there are 12 spheres which perfectly fit in the cuboid in $$3 \times 2 \times 2$$ orientation. All the spheres are identical. The dimensions of the cuboid $$(l:b:h)$$ are in $$3:2:2$$ ratio. The cuboid has an inner volume $$V_c$$, the sphere has an outer volume $$V_s$$, inner volume $$v_s$$, outer radius $$R$$, inner radius $$r$$ and width $$w$$. The volume of the empty spaces in the cuboid (i.e. difference between the volume of the cuboid and the volumes of all the spheres) is equal to the inner volume of 30 such spheres. If the outer radius of a sphere is $$x$$, find $$r$$ in terms of $$x$$.

If for positive integers $$a,b,c,d,e$$, we have $$r = x \left (\sqrt[e]{ \frac{a}{b \pi} - \frac{c}{d} } \right)$$, where $$\text{gcd}(a, b)=\text{gcd}(c,d)=1$$, with $$e$$ minimized. Find the value of $$a+b+c+d+e$$

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