More Cylinder Problems, 5

Category: Solid Geometry

"Published in Newark, California, USA"

An indoor roller skating rink with an area of 1,500 sq. yd. has a concrete flooring 3 in. thick. Find the amount of concrete used in laying the floor.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above 
 

Photo by Math Principles in Everyday Life

The floor for the roller skating rink must be smooth and uniform in thickness. If the thickness of the floor is the same within the roller skating rink, then it is considered a cylinder.
Since the surface area of a roller skating rink as well as the thickness are given in the problem, then we can get the amount of concrete which is the volume of a cylinder used in laying the floor as follows

Therefore, the amount of concrete required for laying the floor is

 

More Cylinder Problems, 4

Category: Solid Geometry

"Published in Newark, California, USA"

The average depth of a lake is estimated to be 40 ft. If the surface area is 15 acres, find the volume of water in the lake. (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 lake is in a shape of a cylinder. Since the surface area of a lake as well as the estimated depth are given in the problem, then we can get the volume of water in the lake as follows

Therefore, the volume of water in the lake is

 

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,