## 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