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'''By: Tao (Steven) Zheng'''
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A fun problem I came up in second year at UBC.
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== '''Problem''' ==
 
== '''Problem''' ==
 
[[File:Hole cube.jpg|thumb|233x233px]]
 
[[File:Hole cube.jpg|thumb|233x233px]]

Revision as of 02:11, 12 July 2019

By: Tao (Steven) Zheng

A fun problem I came up in second year at UBC.

Problem

Hole cube

Imagine a cube with hole drilled through the centers of each face. Each hole is equal in size. We will call this solid the Holey Cube.

Part 1: If the width of the cube is and the radius of each hole is , determine the volume of the Holey Cube.

Part 2: What is the volume if the width of the cube is 10 cm, and the radius of each hole is 3 cm?

Hint: Read these two articles first

Volume of the Bicylinder

Volume of the Tricylinder

Solution

Part 1

The volume of the Holey Cube is not that simple to determine. It requires a bit of set theory, more specifically the inclusion-exclusion principle.

When the holes are drilled through the center of each face, we can see three cylinders taken out. But wait! What about the intersections of the cylindrical cavities? Since there are three cylinders (A, B, C), there are three bicylinders (AB, AC, BC) to patch up. But wait! What about the intersection of all three cylinders? Finally we need to remove a tricylinder (ABC).

Hence, the volume of the Holey Cube is

.

Part 2

If

Failed to parse (syntax error): {\displaystyle s = 10 \: cm } and

Failed to parse (syntax error): {\displaystyle r = 3 \: cm } ,

Failed to parse (syntax error): {\displaystyle V = 457.2401 \: {cm}^{3} } .