Final answer:
To find the area of one base of the right pentagonal prism, you calculate the lateral surface area and subtract it from the total surface area, then divide the result by two, resulting in an area of 25.6 cm² for one base.
Step-by-step explanation:
To determine the area of one of the bases of the right pentagonal prism with a given surface area and height-to-side length ratio, we can follow these steps:
- Firstly, calculate the area of the lateral faces by using the prism height and perimeter of the base.
- Then, subtract the lateral area from the total surface area to find the combined area of the two pentagonal bases.
- Finally, divide this result by two to find the area of one base.
We are given that the height of the prism is ⅓ the side of the base, and the perimeter of the base is 45 cm. With a regular pentagon, each side is 45 cm ÷ 5 = 9 cm. Hence, the height of the prism is 9 cm × ⅓ = 3 cm.
The lateral surface area can be calculated as the perimeter of the base times the height of the prism (L = perimeter × height), which is 45 cm × 3 cm = 135 cm².
To find the total area of the two bases, subtract the lateral area from the total surface area: 186.2 cm² - 135 cm² = 51.2 cm².
Divide this by two to obtain the area of one base: 51.2 cm² ÷ 2 = 25.6 cm².