Melba claimed that she could slice a pentagonal prism perpindicular to the base to create two separate prisms. What is the fewest number of surfacesthat each of the resulting prisms could have?
Five, counting the two ends. The other piece would have six.
To determine the fewest number of surfaces each resulting prism could have, we need to analyze the original pentagonal prism and the process of cutting it.
A pentagonal prism has three distinct types of surfaces:
1. The bases - There are two pentagons that form the top and bottom faces of the prism.
2. The lateral faces - There are five rectangular faces that connect the corresponding edges of the two pentagons.
3. The edges - There are ten line segments that connect the vertices of the two pentagons.
When Melba claims she can slice the pentagonal prism perpendicular to the base, we need to understand what she means. If we imagine slicing the prism in a way that separates it into two distinct prisms, we have a few possibilities:
1. Slicing through the midpoint of a lateral face: In this case, each resulting prism would have a base, two lateral faces, and two edges. Therefore, each prism would have a total of 5 surfaces (2 bases + 2 lateral faces + 1 edge).
2. Slicing through an edge: In this case, each resulting prism would have a base, two lateral faces, and three edges. Therefore, each prism would have a total of 6 surfaces (2 bases + 2 lateral faces + 2 edges).
3. Slicing through a vertex: In this case, each resulting prism would have two bases, three lateral faces, and two edges. Therefore, each prism would have a total of 7 surfaces (2 bases + 3 lateral faces + 2 edges).
Therefore, the fewest number of surfaces that each resulting prism could have is 5.