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
 

 

 

 

 

Right Circular Cylinder Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

Find the amount of waste in making the largest possible cylindrical rod from a bar of iron 3 ft. long which has a square cross section whose diagonal is 6 in.

Photo by Math Principles in Everyday Life

Solution:

To analyze the given word problem, let's label further the given figure as follows

Photo by Math Principles in Everyday Life

The first that we need to do is to solve for the area of the cross section of the waste material. The cross section of the waste material is the difference of a square and a circle. 

By using Pythagorean Theorem, the length of one side of a square which is also the diameter of a circle is

Hence, the area of the cross section of the waste material is

 

 

 

 

 

 

 

Therefore, the amount of waste material removed from a bar of iron which is the difference of the volume of a square prism and a right circular cylinder is
 

 

 

                       or

 

 

More Cylinder Problems, 12

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 isosceles triangle of sides 3 ft., 3 ft., and 2 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 isosceles triangle as follows 

Photo by Math Principles in Everyday Life

The point of intersection of the perpendicular line bisectors of the sides of an isosceles triangle is also the center of a circle. The radius of a circle is equal to the distance from the point of intersection of the perpendicular line bisectors of the sides of an isosceles triangle to its vertex. By using trigonometric functions of right triangles, we can solve for the radius of a circle as follows

 

 

  Hence, the radius of a circle is
 

 

 

Therefore, the volume of a circular cylinder is 

But

Hence, the above equation becomes