Circular Cylinder Problems, 8

Category: Solid Geometry

"Published in Newark, California, USA"

The plumb-line distance from the lowest point of the top of the Leaning Tower of Pisa to the ground is 179 ft. The tower leans so that this plumb line strikes the ground 14 ft. from the nearest point of the base as shown. (a) At what angle is its axis now inclined from the vertical? (b) At what angle would its axis have to be inclined from the vertical before it would topple over?

Photo by Math Principles in Everyday Life

Solution:

(a) To understand more the problem, let's label further the figure as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem, the length of the Leaning Tower of Pisa is

Therefore, the angle of inclination with respect to vertical axis for the Leaning Tower of Pisa is

                              or

(b) If the Leaning Tower of Pisa is inclined from the vertical before it would topple over, then the figure will be like this
 

Photo by Math Principles in Everyday Life

Therefore, the angle of inclination with respect to vertical axis for the Leaning Tower of Pisa before it would topple over 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, 28

Category: Solid Geometry

"Published in Newark, California, USA"

Using the vertices of a 9-in. square as centers, and radii equal to 3 in., four arcs are described outside the square. If the figure thus formed is the uniform cross section of cylinder of element 7 in., find the volume and total area of the cylinder.

Solution:

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

Photo by Math Principles in Everyday Life

As you can see from the figure above that there are four three-fourth circles. Hence, the area of the shaded region which is also the base of a right cylinder is 

Therefore, the volume of a right cylinder is

Next, we need to get the perimeter of the base of the shaded region which is equal to the sum of the circumference of four three-fourth circles and the length of the four line segments between the arcs of three-fourth circles as follows 

Hence, the surface or lateral area of a right cylinder is 

Therefore, the total area of a circular cylinder is