Chapter 14.8, Problem 29WE

a.

To determine

To write the number of planes of symmetry and the number of axes of the rotation the solid has.

Answer to Problem 29WE

A right circular cone has many planes of symmetry containing the vertical axis AOand has only one axis of rotation that is the vertical axis AO .

Explanation of Solution

Given information:

The given figure is right circular cone.

The figure ofa right circular cone.

  

A right circular cone has many planes of symmetry containing the vertical axis AO as shown in figure because for each plane which contains AO the figure is symmetric. And the right circular cone has only one axis of rotation, the axis is AO because this is the only axis on which the rotation of the figure is symmetric.

b.

To determine

To write the number of planes of symmetry and the number of axes of the rotation the solid has.

Answer to Problem 29WE

A cube has nine planes of symmetry and seven axes of rotation.

Explanation of Solution

Given information:

The given figure is a cube.

The figure ofa cube.

  

A cube has nine planes of symmetry. Assume the centroid of the cube is at the origin. All three principal planes are planes of symmetry.

Plane containing side BC and EF is a plane of symmetry.

Plane containing side AD and GH is a plane of symmetry.

Plane containing side AF and HC is a plane of symmetry.

Plane containing side ED and GB is a plane of symmetry.

Plane containing side AB and EH is a plane of symmetry.

Plane containing side FG and DC is a plane of symmetry.

A cube has seven axes of rotation.

Since it is assumed thatthe centroid of the cube is at the origin. Hence, all three principal axes are the axes of rotation because the figure is symmetric about all three principal axes.

Line passing through G and D is an axis of rotation.

Line passing through F and C is an axis of rotation.

Line passing through E and B is an axis of rotation.

Line passing through A and H is an axis of rotation.

c.

To determine

To draw the figure if there is one that meets the conditions.

Answer to Problem 29WE

A trapezoid has six planes of symmetry and four axe of rotation.

Explanation of Solution

Given information:

The given condition is that a parallelogram with just one symmetry line.

The figure of a trapezoid.

  

The above figure has six planes of symmetry. Each plane contains a side of any triangle of the trapezoid and bisects the triangle opposite to that side.

The figure has four axes of rotationeach passes from a vertex of trapezoid and also passes from the centre of opposite triangle.