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This problem trains for: AMC-8, AMC-10, Math Kangaroo 9-10, SAT-II.
A box with side lengths of 3, 5 and 9 units is cut by a plane through three points that are on the edges of the box, each 2 units away from a given vertex V. The portion of the solid that V belongs to is then placed upon a plane with the vertex V pointing upwards. How high above the plane does V reach?
The cut produces a right tetrahedron VABC, as in the figure:
The edges AB, BC and AC have the length:
Therefore, the triangle ABC is equilateral.
The volume of the pyramid VABC can be calculated in two ways:
Where H is the height from V to ABC.
Solve for the height:
The area of the equilateral triangle is:
Substitute the area in the formula for the height: