You are making poster board boundary markers in the shape of square pyramids for a relay race. If you are including a 10-inch square as the bottom and you want the slant height to be 12 inches, how much poster board do you need?
A: 340 inch
B: 360 inch
C: 384 inch
D: 400 inch
To calculate the amount of poster board needed, we need to find the total surface area of the square pyramid.
First, let's find the area of the base, which is a 10-inch square. The area of a square is given by the formula A = side * side. Therefore, the area of the base is 10 * 10 = 100 square inches.
Next, let's find the area of each triangular face. Each face is an isosceles triangle with a base equal to the side length of the square (10 inches) and a slant height of 12 inches. The formula for the area of an isosceles triangle is A = (base * height) / 2. We need to find the height of the triangle using the Pythagorean theorem. The height (h) can be calculated using the formula h = sqrt(slant height^2 - (base/2)^2). Substituting the given values, we have h = sqrt(12^2 - (10/2)^2) = sqrt(144 - 25) = sqrt(119) ≈ 10.92 inches. Now we can calculate the area of each triangular face as A = (10 * 10.92) / 2 ≈ 54.6 square inches.
Since the square pyramid has 4 triangular faces, the total area of the triangular faces is 4 * 54.6 = 218.4 square inches.
Finally, to find the total area, we need to add the area of the base and the area of the triangular faces. Therefore, the total surface area is 100 + 218.4 = 318.4 square inches.
However, we need to make sure that we have enough poster board to cover the entire surface of the pyramid, including the inside edges. To account for this, we multiply the total surface area by 1.1 (10% extra) to ensure sufficient coverage. Thus, the total amount of poster board needed is 318.4 * 1.1 ≈ 350.24 square inches.
Since the question asks for the amount of poster board needed in inches, we can round up to the nearest whole number. Therefore, the answer is approximately 351 inches. None of the given answer choices match exactly, but the closest option is 340 inches, so the correct answer would be A: 340 inch.