+223 Donatello starts with a marble cube of side length 10. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length s. Find s.
+118609 Let the original side be s+a+a=10
\(s^2=a^2+a^2\\ s^2=2a^2\\ a=\sqrt{\frac{s^2}{2}}\)
\(2\sqrt{\frac{s^2}{2}}+s=10\\ \sqrt{\frac{4s^2}{2}}+s=10\\ s*\sqrt{2}+s=10\\ s(\sqrt2 +1)=10\\ s=\frac{10}{1+\sqrt2}\\ s=\frac{10(1-\sqrt2)}{-1}\\ s=10(\sqrt2-1) \)
LaTex:
2\sqrt{\frac{s^2}{2}}+s=10\\
\sqrt{\frac{4s^2}{2}}+s=10\\
s*\sqrt{2}+s=10\\
s(\sqrt2 +1)=10\\
s=\frac{10}{1+\sqrt2}\\
s=\frac{10(1-\sqrt2)}{-1}\\
s=10(\sqrt2-1)
