Final answer:
The total surface area of the hexagonal prism is 126 cm². The square's apothem and radius lengths are both approximately 12.73 inches.
Step-by-step explanation:
To find the total surface area of a hexagonal prism, we need to add the areas of all of its faces. This prism has 6 lateral faces and 2 hexagonal bases. We're given that each base has an area of 15 cm² and each lateral face has an area of 16 cm².
So for the bases, we add: 15 cm² + 15 cm² = 30 cm².
For the lateral faces, we multiply the area of one face by 6: 16 cm² * 6 = 96 cm².
Lastly, we add the areas of the bases and the lateral faces to find the total surface area: 30 cm² + 96 cm² = 126 cm².
For the apothem and radius of a square, remember that the apothem of a square bisects one of the sides creating two right triangles. This means that the apothem is half the length of the diagonal of the square. Through the Pythagorean Theorem, we can find that the diagonal of a square with sides of length 18 inches is roughly 25.46 inches. Divide this by 2 to find the apothem, roughly 12.73 inches. The radius of a square is simply half the length of the diagonal, so the radius is also approximately 12.73 inches.
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