More Prism Problems

Category: Solid Geometry

"Published in Newark, California, USA"

A football stand 150 yd. long has 20 tiers of seats; each tier has a rise of 2 ft. and tread of 3 ft. It is constructed of reinforced concrete with a cross section as shown. Find the amount of material used in its construction.

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Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

The first thing that we need to do is to isolate the given figure which is the 20 tiers of seat section as follows

Photo by Math Principles in Everyday Life

At the first layer, there is one block whose dimension is 2' x 3'. At the second layer, there are two blocks. At the third layer, there are three blocks, and so on. We need to get the total number of blocks so that we can get the area of the section. Since there's a sequence of number of blocks for each layer by arithmetic progression, then we can get the total number of blocks by using the sum of n terms as follows

 

 

 

   
The total area of blocks is
 

 

 

 

   
The area of a triangle is
 

 

 

   
Hence, the area of the 20 tiers of seat section which is the base of a prism is
 

 

 

 
Therefore, the amount of material used for the construction of a football stand which is a prism is
 

 

 

 

 

Trapezoid Prism Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

(a) Find the volume of water in a swimming pool with vertical ends and sides. The length measured at the water line is 50 ft. and the breadth is 20 ft. The bottom of the swimming pool is a plane slopping gradually downward so that the depth of the water at one end is 4 ft. and at the other end is 8 ft.
(b) If the sides, ends, and bottom of the swimming pool are constructed of tile blocks whose glazed surface dimensions are 3 in. by 6 in., and if the ends and sides of the pool  extend 2 ft. above the water level, find the number of blocks used if 1/20 of the surface area is covered by sealing material.

Solution:

(a) To illustrate the problem, it is better to draw the figure as follow

Photo by Math Principles in Everyday Life

Did you notice that the swimming pool is a trapezoid prism whose two opposite faces are right trapezoid? Well, the area of the base which is a right trapezoid is

Therefore, the volume of water in a swimming pool which is a trapezoid prism is

 

 
(b) If the ends and sides of the pool extend 2 ft. above the water level, the dimensions of the pool are as follows

Photo by Math Principles in Everyday Life

We need to get the unknown dimensions of the sides of the pool first so that we can solve for the total surface area of the pool especially the area of the bottom of the pool. Let's analyze a surface of the pool as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem,

Hence, the total surface area of the pool is

 

 

 

   
The total area of the sealing material used is
 

 

 

   
The total area of the tile blocks used is
 

 

 

   
The area of a tile block is
 

 

 

 

 

   
Therefore, the number of tile blocks used for the pool is
 

 

 

   
The number of tile blocks must be in whole numbers. We have to round off to the nearest ones. The answer is