Final answer:
The perimeter of each hexagonal base on a regular hexagonal prism with a volume of 350 cubic inches and a height of 14 inches is approximately 17.34 inches.
Step-by-step explanation:
To find the perimeter of each of the hexagonal bases of a regular hexagonal prism, we will first need to know the area of the hexagonal base. This can be obtained by dividing the volume of the prism by its height. The given volume of the hexagonal prism is 350 cubic inches, and the given height is 14 inches.
The formula for the volume of a prism is Volume = Base Area × Height. Using the given values, we can solve for the base area:
- Base Area = Volume / Height
- Base Area = 350 in³ / 14 in
- Base Area = 25 in³
The formula for the area of a regular hexagon is Area = (3√3)/2 × side length². Let's denote the side length as 's'.
We set up an equation and solve for 's':
- 25 in³ = (3√3)/2 × s²
- s² = 2 × 25 in³ / (3√3)
- s ≈ 2.89 in (After calculation)
Now that we know the side length of the hexagon, we can find the perimeter.
- Perimeter of hexagon = 6 × side length
- Perimeter of hexagon = 6 × 2.89 in
- Perimeter of hexagon ≈ 17.34 in
Thus, the perimeter of each hexagonal base is approximately 17.34 inches.