Triangular Prism Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

A trough is formed by nailing together, edge to edge, two boards 11 ft. in length, so that the right section is a right triangle. If 15 gals. of water are poured into the trough and if the trough is held level so that a right section of the water is an isosceles right triangle, how deep is the water? (231 cu. in. = 1 gal.)

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Solution:

Consider the cross section of a trough which is the base of a triangular prism

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The cross section area of water in a trough is

The length of a trough in inches is

The volume of water in a trough in cubic inches is

Therefore, the height or deep of water in a trough is

Trapezoid Prism Problems

Category: Solid Geometry

"Published in Vacaville, California, USA"

A trench is 180 ft. long and 12 ft. deep, 7 ft. wide at the top and 4 ft. at the bottom. How many cubic yards of earth have been removed?

Solution:

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

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The amount of earth removed is in the shape of a prism whose base is a trapezoid. The area of the base which is a trapezoid is

The amount of earth removed which is the volume of a trapezoid prism is

Therefore, the amount of earth removed in cubic yards is

Polygon Prism Problems, 3

Category: Solid Geometry

"Published in Vacaville, California, USA"

One part of a quartz crystal is a hexagonal prism with a right section of 1.29 sq. in., an edge of 2.31 in., and a base of 1.41 sq. in. Find the altitude of the prism.

Solution:

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

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The volume of a quartz crystal is

Therefore, the altitude of a quartz crystal is

Polygon Prism Problems, 2

Category: Solid Geometry

"Published in Vacaville, California, USA"

The lateral surface of a concrete octagonal pier of height 10 ft. is to be resurfaced. If each of the base edges is 1 ft., how many square feet of surfacing will be required?

Solution:

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

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The lateral surface area of a concrete octagonal pier required to be resurfaced is  

 

Polygon Prism Problems

Category: Solid Geometry

"Published in Newark, California, USA"

A lead pencil whose ends are regular hexagons was cut from a cylindrical piece of wood, with the least waste. If the original piece was 8 in. long and ½ in. in diameter, find the volume of the pencil.

Solution:

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

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The first thing that we need to do is to consider the base of the pencil which is a regular hexagon and draw three diagonals that passes thru its center as follows

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Next, we need to consider a triangle in a regular hexagon so that we can solve for its area as follows

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Since all the interior angles of a triangle are congruent, then it is an equilateral or equiangular triangle. All sides of an equilateral triangle are congruent. The base of a triangle is also equal to ¼ in.

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The area of a triangle is

Hence, the area of the base of a regular hexagon is

Therefore, the volume of a pencil which is a regular hexagon prism is  

or