More Cylinder Problems, 9

Category: Solid Geometry

"Published in Newark, California, USA"

A cylinder whose base is a circle is circumscribed about a right prism of altitude 12.6 ft. Find the volume of the cylinder if the base of the prism is a square of edge 3 ft. 

Solution:

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

Photo by Math Principles in Everyday Life

Consider the base of a cylinder which is a circle that circumscribed about a square as follows

Photo by Math Principles in Everyday Life

The length of the diagonal of a square is equal to the diameter of a circle. By Pythagorean Theorem, the diameter of a circle is

Therefore, the volume of a circular cylinder is

But
 

Hence, the above equation becomes
 

 

 

or

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,