To find the height of the prism, we need to use the formula for the volume of a prism:
Volume = Area of base Height
We have two potential heights given, which are 12 mm and 8 mm. We have the volume of the prism given as 4,560 cubic millimeters. We need to determine which height will give us this volume with a positive base area.
We can check each potential height to see if it yields the correct volume:
For 12 mm height:
Volume = Area of base Height
4,560 mm³ = Area of base 12 mm
Area of base = 4,560 mm³ / 12 mm
Area of base = 380 mm²
For 8 mm height:
Volume = Area of base Height
4,560 mm³ = Area of base * 8 mm
Area of base = 4,560 mm³ / 8 mm
Area of base = 570 mm²
Now, since we are provided with the volume and potential heights only (and considering the volume has a fixed value), the correct height will be the one that matches this volume with a possible real positive area of the base. Both calculated base areas are positive, which is possible for a real base of a prism.
However, we are not provided with an actual way to validate which height is correct based on the base area, because a regular pentagon can have many possible side lengths and corresponding apothem lengths (the perpendicular distance from the center to a side) that result in different base areas.
Given the information, we cannot conclusively determine which height is correct solely with the volume, as both calculations for base area result in positive values. Therefore, we lack sufficient information to provide a definitive height.
In the context of this problem, without additional information (such as the length of a side of the pentagonal base or its apothem), both heights yield feasible base areas. However, since we must choose one height and the exercise seems to assume only one correct height, we would generally check for any additional context or clues.
Since we don't have such context or clues in the provided information, we cannot determine the correct height from 12 mm and 8 mm given the volume alone. Hence, the problem seems incomplete, and an assumption must be made or additional base properties must be supplied to obtain a definite answer.
