More Cylinder Problems, 8

Category: Solid Geometry

"Published in Newark, California, USA"

Find the volume of the largest cylinder with circular base that can be inscribed in a cube whose volume is 27 cu. in.

Solution:

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

Photo by Math Principles in Everyday Life

In this problem, the volume of a cube is given. The length of the edge of a cube is

          

             

     
If the circular arc of the base of a cylinder is tangent to the bottom edges of a cube, then it follows that
        

     
where x is the length of the edge of a cube and r is the radius of a circle. The radius of a circle is
  

        

         

          
Therefore, the volume of the largest cylinder that can be inscribed in a cube is
          

          
But
          

      
Hence, the above equation becomes,
      

           

                   or

         

            

More Cylinder Problems, 7

Category: Solid Geometry

"Published in Newark, California, USA"

The crown of a straw hat has a base of 38 sq. in. The depth of the crown is 3 in. (Inside dimensions are given.) If the head occupies two-thirds of the space enclosed by the crown, find the volume remaining for ventilation. 

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

Since the area of the base or crown of a straw hat as well as the depth of the crown are already given, then we can get the volume of the crown of a straw hat as follows
 

 

 

 
If the head occupies two-thirds of the space or volume of the crown, then the volume remaining for the ventilation is one-third of the space of the crown.
 
Therefore,
 

 

 

More Cylinder Problems, 6

Category: Solid Geometry

"Published in Newark, California, USA"

The outer protective smokestack of a steamship is streamlined so that it has a uniform oval section parallel to the deck. The area of this oval section is 48 sq. ft. If the length of the stack is 15 ft. and the stack is raked aft so that its axis at its upper end is horizontally 4.2 ft. from the lower end, find the volume enclosed by the stack. (Raked means inclined.)

Solution:

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

Photo by Math Principles in Everyday Life

The area of a base which is an oval section is already given in the problem. By Pythagorean Theorem, the altitude of the smokestack is

Therefore, the volume of a smokestack which is a cylinder is