a. Can a polyhedron have 19 faces, 34 edges, and 18 vertices? Explain how you kn

Question

a. Can a polyhedron have 19 faces, 34 edges, and 18 vertices? Explain how you know.

b. Is a cone a polyhedron? Explain how you know.

c. How many edges and vertices are there for an octahedron, which is a polyhedron with eight congruent triangular faces? Describe your steps for answering this question.

d. What are the formulas for the volumes of a sphere, a cone with a height equal to its radius, and a cylinder with its height equal to its radius? How are these formulas related?

e. A classmate says that a rectangular prism that is 6 cm long, 8 cm wide, and 15 cm high is similar to a rectangular prism that is 12 cm long, 14 cm wide, and 21 cm high. Explain your classmate's error.

f. A medium sized box an hold 55 T-shirts. If the dimensions of a jumbo box are three times that of the medium box, how many T-shirts can the jumbo box hold? Explain.

Explanation / Answer

A) No because if use Euler's formula i.e. F+V = E+2, the equation would not work because 19+18 is not equal to 34+2.

So the given polyhedron is not possible.

B) A Polyhedron is a solid with flat faces only. So a Cone is not a polyhedron as it has curved surfaces.

C) we have an octahedron, which is a polyhedron with eight congruent triangular faces. Each pair of faces shares one edge. So number of edges is 8×3/2 = 12 edges.

Each face has three angles and 4 angle meet at each vertex. So number of vertex it has is 8×3/4 = 6 vertex.

D) volume of a Sphere = (4/3)?r^3........(1)

Volume of cone = (1/3) ? (r^2)h

If we have r=h , then volume of cone becomes (1/3)?r^3......(2)

Now

Volume of cylinder = ?(r^2)h

If we have r=h , then volume of cylinder becomes ?r^3..........(3)

By comparing eq (1),(2) and (3), we get

(4/3)Volume of Sphere = volume of cylinder = (1/3)volume of cone

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