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Saturday, May 3, 2014

More Cube Problems, 5

Category: Solid Geometry

"Published in Vacaville, California, USA"

One cube has a face equivalent to the total area of another cube. Find the ratio of their volumes.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Let B = be the area of a base of a large cube
      x = be the length of the edge of a large cube
      T = be the total area of a small cube
      y = be the length of the edge of a small cube
      V1 = be the volume of a large cube
      V2 = be the volume of a small cube 

The area of a base of a large cube is


The total area of a small cube is

 
From the given problem statement, we know that
 
 



The volume of a large cube is


 
 
 

The volume of a small cube is


Therefore, the ratio of their volumes is


 

Friday, May 2, 2014

More Cube Problems, 4

Category: Solid Geometry, Plane Geometry

"Published in Newark, California, USA"

Find the area of a triangle whose vertex is at the midpoint of an upper edge of a cube of edge a and whose base coincides with the diagonally opposite edge of the cube.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The first thing that we need to do is to solve for the altitude of a triangle first. Since one of the vertex of a triangle is located at the midpoint of the upper edge of a cube, then a triangle is an isosceles triangle. The altitude of an isosceles triangle is equal to the hypotenuse of an isosceles right triangle at the right side of a cube. 

By Pythagorean Theorem






Therefore, the area of a triangle is




Thursday, May 1, 2014

Cube - Prism Problems

Category: Solid Geometry

"Published in Newark, California, USA"

A vegetable bin built in the form of a cube with an edge of 6 ft. is divided by a vertical partition which passes through two diagonally opposite edges. Find the lateral surface of either compartment.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

If a plane or a divider connects the two opposite sides of a cube, then a cube becomes two triangular prisms with equal bases, lateral surfaces, and their volumes as well.

If the sides of a cube are all equal, then the bases of two prisms are all isosceles right triangles.

Use Pythagorean Theorem in order to solve the another side of the base of a triangular prism which is the hypotenuse as follows







Therefore, the lateral surface area of a triangular prism which is a compartment of a vegetable bin is




or