Final answer:
The length of the game piece is 27.5 mm, calculated by using the perimeter to find the side length of the square base, and then applying the Pythagorean theorem to determine the height of one pyramid and doubling it since the piece consists of two identical pyramids.
Step-by-step explanation:
To find the length of the game piece shaped like two identical square pyramids attached at their bases, we first need to determine the side length of the square base using the perimeter. The perimeter of the square base is given as 80 millimeters. Since the perimeter of a square is the sum of all four sides, we divide this number by 4 to find the length of one side:
Side length of the square base = Perimeter ÷ 4 = 80 mm ÷ 4 = 20 mm.
The question asks for the length of the game piece, which will be the total height of the two pyramids when attached at their bases. Since the pyramids are identical, we only need to find the height of one pyramid and then double it. The slant height of 17 mm will not help us find the height directly because it is the length from the apex of the pyramid to the mid-point of a side of the square base, not the perpendicular height we need. We must use the Pythagorean theorem to find the perpendicular height (h):
h2 + 102 = 172
h2 + 100 = 289
h2 = 189
h = √189 ≈ 13.75 mm (height of one pyramid).
Therefore, the length of the game piece is twice this height:
Length of the game piece = 2 × 13.75 mm = 27.5 mm.