More Right Circular Cylinder Problem, 2

Category: Solid Geometry

"Published in Newark, California, USA"

A right cylindrical solid of altitude 6 in. has the cross section shown in the shaded portion of the figure. BEDG is a circle whose radius is OG. AFCG is a circle which is tangent to the larger circle at G. If AB = CD = 5 in. and EF = 9 in., find the volume of the cylinder.

Photo by Math Principles in Everyday Life

Solution:

The cross section of a right circular cylinder consists of two tangent circles with their common point at Point G. Let's further analyze and label the cross section of a right circular cylinder as follows

Photo by Math Principles in Everyday Life

We noticed that line segments AC and FG are the chords of a small circle that intersect at point O, which is also the center of a big circle. From Plane Geometry, we know that the product of two divided chords is equal to the product of other two divided chords. We can solve for the radius of a big circle as follows

The line segment OF is calculated as follows

The radius of a small circle is calculated as follows

The area of the shaded portion of the cross section of a right circular cylinder is calculated as follows

             Area of a base = Area of Big Circle - Area of Small Circle

Therefore, the volume of a right circular cylinder is

or

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, 22

Category: Solid Geometry

"Published in Vacaville, California, USA"

A cylindrical tank, diameter 1 ft., length 6 ft., is placed so that its axis is horizontal. How many pounds of water will be used in filling it to a depth of 9 in., if water weighs 62.4 lb. per cu. ft.?

Solution:

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

Photo by Math Principles in Everyday Life

If a right circular cylinder whose axis is placed in a horizontal position, then its base is a circular segment. We need to get the area of its base first so that we can solve for the volume of water in a right circular cylinder. Let's analyze the cross section which is the base of a cylinder that is partially filled with water as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem, the length of AD or DB is

Since ∆AOD or ∆BOD is a right triangle, then by using trigonometric function, the value of ∠AOD or ∠BOC is

The area of circular segment ABC is

The area of the base of a cylinder filled with water is

                          or

Hence, the volume of water in a right circular cylinder is

Therefore, the weight of water in a right circular cylinder is