¶ 2018 AMC 10B Problem 10
Problem:
In the rectangular parallelepiped shown, , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?
Answer Choices:
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Solution:
The volume of the rectangular pyramid with base and apex equals the volume of the given rectangular parallelepiped, which is , minus the combined volume of triangular prism , tetrahedron , and tetrahedron . Tetrahedra and each have three right angles at and , respectively, and the edges of the tetrahedra emanating from and have lengths , and , so the volume of each of these tetrahedra
is . The volume of the triangluar prism is because it is half the volume of the rectangular parallelepiped. Therefore the requested volume is .
Let and be the midpoints of and , respectively. By the Pythagorean Theorem . Let be the foot of the perpendicular from to . Then , so
The requested volume of the pyramid is times the area of the base times the height, which is
The problems on this page are the property of the MAA's American Mathematics Competitions