Question

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 $s.$ [asy] import three; size(7cm); unitsize(1 cm); currentprojection

Answer 1

Answer:

The length of each side of the polyhedron, s is 4.142 or

= $10(\sqrt{2}-1)$

Step-by-step explanation:

If we take the distance of the length of the side of the cube where Donatello make the cut to the corner of the cube as x we get the side of length each polyhedron as

$\sqrt{x^2 + x^2} = \sqrt{2x^2} =x\sqrt{2}$

x·√2 because we have an hy

Therefore, the original cube had sides of length

10 = 2·x + x·√2

Solving for x we get

$x = \frac{10}{\sqrt{2}+2 }$

Therefore the length of s is x·√2 which gives

$s = x\sqrt{2} = \frac{10}{\sqrt{2}+2 }\sqrt{2} = 10\sqrt{2} -10 = 10(\sqrt{2}-1) = 4.142$

The length of each side of the polyhedron, s = 4.142.