Crazy coordinates

Geometry Level pending

A polyhedron has the following vertices, for some real number, a a :

  • ( e , π , a ) (e,\pi,a)
  • ( π , e , a ) (\pi,e,a)
  • ( − π e , 8 , a ) (-\pi e,8,a)
  • ( e 2 + π , − π , a ) (e^2+\pi,-\pi,a)
  • ( − π , − π 3 , a ) (-\pi,-\pi^3,a)
  • ( π , e , a + π ) (\pi, e, a+\pi)

And it has a volume of 20 20 .

What would be its volume if the last vertex were changed to this?

  • ( 6 π , π + 3 e , a + 8 π ) (6\pi, \pi + 3e, a+8\pi)

If you think that it can't be determined, put -1 as your answer.


The answer is 160.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Apr 8, 2017

Since the first five points share the same z z coordinate, then the volume will simply be proportional to how far the z z component of this final vertex is from this value, a a , and it is independent of its x x and y y coordinates.

Since V ( z = a + π ) = 20 V(z = a+\pi) = 20 , then

V ( z = a + 8 π ) = 160 V(z = a+8\pi) = \boxed{160}

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