More Prism Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

Show that in any prism the area of a right section is equal to the product of the base and the sine of the angle between a lateral edge and the base. (This relation is used in connection with the reflection of light through prisms.)

Solution:

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

Photo by Math Principles in Everyday Life

If the area of a base and the altitude are given, then the volume of a prism is

If the area of a right section and the length of a lateral edge or slant height are given, then the volume of a prism is

Equate the two equations above, we have

Consider a right triangle from the figure above,

Substitute the value of h to the above equation, we have

Therefore,

More Cylinder Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

An ice-storage plant removed from the center of a pond a mass of ice covering an area of 2 acres. If the ice had a uniform thickness of 2 ft., find the weight in tons of the ice removed. (Ice weighs 56 lbs. per cu. ft.; 1 ton = 2,240 lbs.; 1 acre = 43,560 sq. ft.)

Solution:

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

Photo by Math Principles in Everyday Life

The area of the base which is the top of ice covering from the center of a pond is

The volume of an ice covering which is a cylinder is

The weight of an ice covering is

Therefore, the weight of an ice covering in metric tons is

More Cylinder Problems, 2

Category: Solid Geometry

"Published in Newark, California, USA"

During a rain, ¼ in. of water fell. Find how many gallons of water fell on a level 10-acre park. (Take 1 cu. ft. = 7.48 gals. and 1 acre = 43,560 sq. ft.)

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

Let's assume that the thickness of the rainwater within the level 10-acre park is ¼ in. Since the thickness of the rainwater is uniform, then the given figure is considered as a cylinder. 

In this case, the area of the base which is the level 10-acre park in square feet is

The volume of the rainwater which is a cylinder is

Therefore, the volume of the rainwater in gallons is