Final answer:
No, a (genus-0) polyhedron with exactly 4 facets, each being a 5-gon, is not possible.
Step-by-step explanation:
A (genus-0) polyhedron with exactly 4 facets, each being a 5-gon, is not possible.
A polyhedron is a solid figure with flat faces. In order to determine the number of facets on a polyhedron, we use Euler's formula: F + V - E = 2, where F represents faces, V represents vertices, and E represents edges.
For a (genus-0) polyhedron with 4 facets, each being a 5-gon, we have F = 4.
Since a 5-gon has 5 sides, each facet will have 5 edges. Therefore, E = 4 x 5 = 20.
Using Euler's formula, we can rearrange it to solve for V. So, V = 2 - F + E = 2 - 4 + 20 = 18.
For a (genus-0) polyhedron, the number of vertices must be greater than or equal to 4. But in this case, we have V = 18. Since V is greater than 4, it means that a (genus-0) polyhedron with exactly 4 facets, each being a 5-gon, is not possible.