Grades 11-12 Video Solutions 2011-11&12 Video Solutions 2011 problem30

Video Transcription

30. A cube with dimensions 3x3x3 is constructed out of 27 smaller cubes with dimensions 1x1x1. So here I have such a cube. The front face and the back face are marked with 3x3 grids  and I have placed a cube here so that one corner is at the origin and the opposite corner in Cartesian coordinates has dimensions 3,3,3. Our question here is to decide how many of  these smaller cubes intersect the plane which passes through the center of the cube and is perpendicular to the line passing through two opposite vertices of the cube. So we have  a normal vector that will be along the line passing through the origin and 3,3,3 in space and we know the center of the cube is the point with coordinates 3,2,3,2,3. So we will  find the plane through the center in the direction normal to the line passing through these two points 0,0,0 the origin and 3,3,3. So this normal vector is just a vector with  terminal point at 3,3,3 and so now we can write down the equation of the plane that is 3x-3 halves plus 3y-3 halves plus 3z-3 halves all equal to zero. We can simplify this divide  by 3 on both sides and this becomes x plus y plus z is equal to 9 halves. Now this tells us that the plane we are considering does not cross through any of the vertices of any  of the 27 smaller cubes because vertices have integer coordinates in the Cartesian coordinate system and it is not possible to add three integers and obtain a nine halves.  So the plane misses all vertices and we then note that if it misses all vertices it will miss here the lower cube if we look at this face straight on this face here will be missed  and so the cube below that or after it and the whole prism made up of three cubes here like that  will be missed and the same thing here on the top face this cube and the three cubes here after it  they will they will be missed so that's six all together and then also the cube whose top face  is here that's another one and the cube whose bottom face is here so that's another one and  these will not intersect the plane all the other ones will so we miss a total of eight cubes  with the plane and the remaining 19 intersect the plane so that is our answer 19 cubes intersect the plane

Video Summary

The problem involves a 3x3x3 cube made of 27 smaller 1x1x1 cubes. A plane passes through the cube's center and is perpendicular to a line between opposite vertices (0,0,0) and (3,3,3), forming the equation x + y + z = 9/2. Since this plane doesn’t cross any cube vertices (which have integer coordinates), it misses 8 cubes completely. Therefore, 19 cubes intersect the plane. Thus, the answer is 19 cubes intersect the plane.

Keywords

3x3x3 cube

intersecting plane

integer coordinates

opposite vertices

19 cubes