Rectangular Parallelepiped Problem, 15

Category: Solid Geometry

"Published in Newark, California, USA"

Two rectangular water tanks with tops on the same level are connected by a pipe through their bottoms. The base of one is 6 in. higher than that of the other. Their dimensions are 4 ft. by 5 ft. by 2½ ft. and 4 ft. by 7 ft. by 3 ft., respectively. How deep is the water in the larger tank when the water they contain equals half their combined capacity, if the 2½ ft. and 3 ft. edges are vertical?

Solution:

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

Photo by Math Principles in Everyday Life

The total volume of two empty rectangular water tanks is

If two tanks are filled with water which is equal to one-half of the total volume of two empty tanks, the above figure becomes

Photo by Math Principles in Everyday Life

Therefore, from the figure above, the depth or height of water in the larger tank is
 

 

 

 

 

 

 

 

 

Trapezoid Prism Problem, 6

Category: Solid Geometry

"Published in Vacaville, California, USA"

A dam is 40 ft. long, 12 ft. high, 7 ft. wide at the bottom, and 4 ft. wide at the top. How many cubic yards of material were used in constructing it?

Solution:

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

Photo by Math Principles in Everyday Life

The area of the base which is the area of the cross section of a trapezoid prism is 

Therefore, the amount of material used in constructing  a dam which is the volume of trapezoid prism is

                           or

Right Circular Cylinder Problems, 16

Category: Solid Geometry

"Published in Vacaville, California, USA"

When an irregular-shaped rock is placed in a cylindrical vessel of water whose radius is 4.18 in., the water rises 6.85 in. What is the volume of the rock if it is completely submerged?

Solution:

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

Photo by Math Principles in Everyday Life

If you have an object with irregular shape, let's say a stone, for example, then it is difficult to get its volume. In this case, the only way to get the volume of an object is by immersion method. You will need a right circular cylinder with water and measure its height. Next, immerse an object in water in a right circular cylinder and measure its height. The difference of their height will be used to calculate the change of the volume of water which is equal to the volume of an object.
     
Since the change of the level of water in a right circular cylinder is given in the problem, therefore, the volume of an object is