A teepee dog bed is in the shape of a triangular pyramid. How much fabric is needed to make the teepee given the following dimensions? The slant height of the equilateral triangular faces is 25 in. The height of the base triangle is 26 in. The base of the equilateral triangles is 30 in. (1 point) Responses 1,365 in.2 1,365 in. squared 1,515 in.2 1,515 in. squared 1,450 in.2 1,450 in. squared 1,495 in.2
To find the total surface area of the teepee dog bed, we need to find the surface area of the two equilateral triangles and the surface area of the three rectangular faces that make up the sides of the pyramid.
The surface area of an equilateral triangle can be calculated using the formula: S = (√3 / 4) * a^2, where a is the side length of the triangle.
Given that the base of the equilateral triangles is 30 in, the height of the equilateral triangles can be found using the Pythagorean theorem:
height = √(25^2 - (30/2)^2) = √(625 - 225) = √400 = 20 in
Therefore, the surface area of one equilateral triangle is:
S = (√3 / 4) * 30 * 20 = 300√3 in^2
Since there are two equilateral triangles making up the sides of the teepee, the total surface area of the equilateral triangles is:
2 * 300√3 = 600√3 in^2
To find the surface area of each rectangular face, we can first calculate the perimeter of the base triangle using the side lengths (30, 30, 30):
Perimeter = 30 + 30 + 30 = 90 in
The height of each rectangular face is the slant height of the equilateral triangle (25 in):
Surface area of one rectangular face = base * height = 30 * 25 = 750 in^2
Since there are three rectangular faces, the total surface area of the three rectangular faces is:
3 * 750 = 2250 in^2
Therefore, the total surface area of the teepee dog bed is:
600√3 + 2250 = 1,365 inches squared
Thus, the correct answer is 1,365 in. squared.