Circular Cylinder Problems, 7

Category: Solid Geometry

"Published in Newark, California, USA"

 An air duct in the form of a circular cylinder has a cross section of diameter 16 in. The distance between the bases is 20 ft., and the elements are inclined at an angle of 50° to the bases. Find the amount of magnesia required to protect the duct with a magnesia covering ½ in. thick.

Solution: 

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

Photo by Math Principles in Everyday Life

Since the distance of the two bases which is the altitude and the angle of inclination of the elements with respect to the bases are given, then we can solve for the length of element which is the length of an air duct by using sine function as follows

The area of a cross section of an air duct is

Therefore, the amount of magnesia required to protect the air duct which is the volume of a circular cylinder is

Circular Cylinder Problems, 6

Category: Solid Geometry

"Published in Newark, California, USA"

Two vertical brine tanks, with tops on the same level, one 16 ft. deep, the other 4ft. deep, have their tops and bottoms connected by pipes 2 in. in diameter. If the pipe connecting the tops measures 5 ft., find the weight of brine in the other pipe when entirely full. (The brine weighs 66.8 lb. per cu. ft.)

Photo by Math Principles in Everyday Life

Solution:

To analyze more the problem, it is better to label further the given figure as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem, the length of a pipe that connects the bottom of two tanks is

 

 

 

 

Hence, the volume of a brine in a pipe that connects the bottom of two tanks is
 

 

 

 

 

 

Therefore, the weight of a brine in a pipe that connects the bottom of two tanks is

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