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

     

Right Circular Cylinder Problems, 18

Category: Solid Geometry

"Published in Newark, California, USA"

How much wood is wasted in turning out the largest possible cylindrical rod from a stick whose uniform square cross-sectional area is 10 sq. in. and whose length is 5 ft.?

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 diameter of a circle is not given. Since the area of a cross-section of a stick is given which is a square, then we can solve for the sides of a square which is also a diameter of a circle as follows

Therefore, the amount of wood wasted in turning out the largest possible cylindrical rod from a stick is