Triangular Prism Problems, 6

Category: Solid Geometry

"Published in Vacaville, California, USA"

In the figure is shown a block of wood in the form of a right prism whose bases are right triangles. A cylindrical auger hole of diameter 2 in. is bored through the block. If the elements of the cylindrical hole are perpendicular to face ABEF and if the lateral surface of the cylindrical hole is tangent to face ABCD, find the volume removed.

Photo by Math Principles in Everyday Life

Solution:

Did you notice that the altitude or length of a right triangular prism is not given in the problem? Well, that's fine because we don't need it in the problem. If the word problem says "If the elements of the cylindrical hole are perpendicular to face ABEF and if the lateral surface of the cylindrical hole is tangent to face ABCD,..", then the side view of the section will be like this
 

Photo by Math Principles in Everyday Life

The darker section is a truncated right circular cylinder because the two bases are not equal and their altitudes or elements are not equal. We have to get the average of their altitudes first before we can solve for the volume of a cylinder. By similar triangles, the other altitude of a cylinder is

 

 

 

Therefore, the amount of a cylindrical auger hole removed from a prism which is the volume of a truncated right circular cylinder is
 

 

 

 

 

More Cylinder Problems, 10

Category: Solid Geometry

"Published in Vacaville, 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 rectangle 3 ft. by 4ft. 

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 rectangle as follows

Photo by Math Principles in Everyday Life

The length of the diagonal of a rectangle 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, 11

Category: Solid Geometry

"Published in Vacaville, 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 an equilateral triangle of side 4 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 an equilateral triangle as follows 

Photo by Math Principles in Everyday Life

The radius of a circle is equal to two thirds of the altitude of an equilateral triangle. By Pythagorean Theorem, the altitude of an equilateral triangle is

Hence, the radius of a circle is

Therefore, the volume of a circular cylinder is  

But

Hence, the above equation becomes

                         or