Circular Cylinder Problems, 5

Category: Solid Geometry

"Published in Vacaville, California, USA"

A channel whose cross section is a semicircle with rise of 1 ft. per 1000 ft., is flowing full. The diameter of the channel is 6.55 ft. The vertical plane which contains the axis is perpendicular to the two vertical planes which contain the ends of the channel. If the end planes are 2000 ft. apart, find the amount of water in the channel.

Solution:

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

Photo by Math Principles in Everyday Life

The first thing that we need to do is to solve for the altitude or vertical distance of the ends of the channel which is a right semicircular cylinder by similar triangles as follows 

By Pythagorean Theorem, the length of the channel is 

Therefore, the amount of water in the channel which is a right semicircular cylinder is

 

Circular Cylinder Problems, 4

Category: Solid Geometry

"Published in Vacaville, California, USA"

A smokestack of a ship is 25 ft. long with a rake aft (angle of the stack's inclination from the vertical) so that its top rises 24 ft. above the deck. The cross section of the flue is a circle with a diameter of 3½ ft. Completely encircling the flue is a protective stack. The distance between the two stacks is 6 in. Find the space enclosed between the two stacks  and also the outside painting surface of the protective stack. (Neglect the thickness of the metal).

Photo by Math Principles in Everyday Life

Solution: 

Consider the given figure above

Photo by Math Principles in Everyday Life

The cross section of a smokestack of a ship is a two concentric circles. The area of the region between the two concentric circles is

Therefore, the amount of space enclosed between the two stacks of a smokestack which is the volume of a circular cylinder is  

and the amount of outside painting the surface of a smokestack is

 

Circular Cylinder Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

A circular concrete conduit, whose inside diameter is 10 ft., is 1 ft. thick. It rises 16 ft. per 1000 horizontal feet. The vertical plane which contains the axis is perpendicular to the two vertical planes which contain the ends of the conduit. If the ends are 3000 ft. apart, find the amount of concrete used in the construction of the conduit.

Solution:

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

Photo by Math Principles in Everyday Life

The first thing that we need to do is to solve for the altitude or vertical distance of the ends of a conduit which is a right circular cylinder by similar triangles as follows

By Pythagorean Theorem, the length of a concrete conduit is

 

 

 

 

The cross section of a concrete conduit is a two concentric circles. The area of the region between the two concentric circles is
 

 

 

Therefore, the amount of a concrete used for the construction of a conduit which is the volume of a circular cylinder is