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

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

More Cube Problems, 3

Category: Solid Geometry, Physics

"Published in Newark, California, USA"

What is the weight of a block of ice 24 in. by 24 in. by 24 in., if ice weighs 92 percent as much as water, and water weighs 62.5 lb. per cu. ft.?

Solution:

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

Photo by Math Principles in Everyday Life

The first that we need to do is to get the volume of an ice which is a cube as follows

Since the density of ice is expressed in lb. per cu.ft., then we need to convert the volume of an ice in cu. ft. as follows

If the specific gravity of ice is 0.92, then the density of ice is

Therefore, the weight of ice is

More Cube Problems, 2

Category: Solid Geometry

"Published in Newark, California, USA"

Plato (429 - 348 B.C.) was one of the first to discover a solution to that famous problem of antiquity, the duplication of a cube, i.e., the finding of the edge of a cube whose volume is double that of a given cube. 
    
One legend asserts that the Athenians, who were suffering from a plague of typhoid fever, consulted the oracle at Delos as to how to stop the plague. Apollo replied that the Delians would have to double the size of his altar, which was in the form of a cube. A new altar was constructed having its edge twice as long as that of the old one. But the pestilence became worse than before, whereupon the Delians appealed to Plato. Given that the side of the altar was 8 ft., find, accurate to five figures, the edge of the required altar.

Solution:

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

Photo by Math Principles in Everyday Life

Consider the first cube or old altar:

Consider the second cube or new altar:

Therefore, the edge of the required or new altar is

More Cube Problems

Category: Solid Geometry

"Published in Vacaville, California, USA"

How much material was used in the manufacture of 24,000 celluloid dice, if each die has an edge of ¼ in.?

Solution:

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

Photo by Math Principles in Everyday Life

The volume of a cube is

Therefore, the amount of material used in making 24,000 celluloid dice is