The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

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:

figure

The edges AB, BC and AC have the length:

equation

Therefore, the triangle ABC is equilateral.

The volume of the pyramid VABC can be calculated in two ways:

equation

Where H is the height from V to ABC.

Solve for the height:

equation

The area of the equilateral triangle is:

equation

Substitute the area in the formula for the height:

equation