38 Review Problem - Circular log with non-uniform cross-section

Problem 38
A log 18 ft. long is 2 ft. in diameter at the top end and 3 ft. in diameter at the butt end.
 

038-circular-log.gif

 

  1. How many cubic feet of wood does the log contain?
  2. How many cubic feet are there in the largest piece of timber of square cross section that can be cut from the log?
  3. How many cubic feet are in the largest piece of square timber of the same size throughout its whole length?
  4. How many board feet does the piece of timber in (c), a board foot being equivalent to a board 1 ft. square and 1 in. thick?
Solids of Revolution by Integration Jhun Vert Tue, 04/28/2020 - 08:27 pm

The solid generated by rotating a plane area about an axis in its plane is called a solid of revolution. The volume of a solid of revolution may be found by the following procedures:
 

The Pyramid Jhun Vert Mon, 04/27/2020 - 10:20 am

A pyramid is a polyhedron with a polygon base of any shape, and all other faces are triangles which have common vertex.
 

oblique-and-right-pyramid.jpg

 

The Cone

Cone
The surface generated by a moving straight line (generator) which always passes through a fixed point (vertex) and always intersects a fixed plane curve (directrix) is called conical surface. Cone is a solid bounded by a conical surface whose directrix is a closed curve, and a plane which cuts all the elements. The conical surface is the lateral area of the cone and the plane which cuts all the elements is the base of the cone.

 

Like pyramids, cones are generally classified according to their bases.
 

Right cone and Oblique Cone with Some Elements

 

The Prism Jhun Vert Mon, 04/27/2020 - 09:17 am

Prism is a polyhedron in which two faces are equal polygons in parallel planes, and all other faces are parallelograms. There are two types of prism; oblique prism and right prism. Oblique prism is when the axis is not perpendicular to the base and right prism if the axis is parallel to the base. In right prism, all lateral areas are rectangle.
 

Elements of a Prism