(a) Can a polyhedron have 10 faces, 20 edges and 15 vertices? Justify your answer.
(b) A glowing bulb is kept above the following solids. Name the shape of the shadows obtained in each case. Attempt to give a rough sketch of the shadow.
[4 MARKS]
(a) Can a polyhedron have 10 faces, 20 edges and 15 vertices? Justify your answer.
(b) A glowing bulb is kept above the following solids. Name the shape of the shadows obtained in each case. Attempt to give a rough sketch of the shadow.
[4 MARKS]
Each part: 2 Marks
(a) By Euler's formula,
F + V - E = 2
Substituting the given values
10 + 15 - 20 = 35 - 20 =15 which is not equal to 2.
A solid needs to satisfy the Euler's formula to be called a polyhedron.
⇒ A polyhedron cannot have 10 faces, 20 edges and 15 vertices.
(b) The shapes of the shadows of these figures will be as follows.
(i) A ball
The shape of the shadow of a ball will be a circle.
(ii) A cylindrical pipe
The shape of the shadow of a circular pipe will be a rectangle.
(iii) A book
The shape of the shadow of a book will be a rectangle.
Each part: 2 Marks
(a) By Euler's formula,
F + V - E = 2
Substituting the given values
10 + 15 - 20 = 35 - 20 =15 which is not equal to 2.
A solid needs to satisfy the Euler's formula to be called a polyhedron.
⇒ A polyhedron cannot have 10 faces, 20 edges and 15 vertices.
(b) The shapes of the shadows of these figures will be as follows.
(i) A ball
The shape of the shadow of a ball will be a circle.
(ii) A cylindrical pipe
The shape of the shadow of a circular pipe will be a rectangle.
(iii) A book
The shape of the shadow of a book will be a rectangle.
