Solar Innovations makes a skylight in the shape of a square pyramid. The pyramid is represented with a net. How many square feet of acrylic are needed to make a skylight covering the lateral surface area? (1 point) ft.2
To calculate the lateral surface area of a square pyramid, we need to find the area of each triangular face and add them together.
Since the pyramid is in the shape of a square pyramid, all the triangular faces are isosceles triangles with a base length equal to the side length of the square base of the pyramid.
Let's assume the side length of the square base of the pyramid is "s".
Each triangular face has a base length of "s" and a height equal to the slant height of the pyramid (let's call it "l").
To find the slant height of the pyramid, we can use the Pythagorean theorem:
l^2 = (s/2)^2 + h^2
l^2 = s^2/4 + h^2
Since the pyramid is a square pyramid, the height of the pyramid, h, is equal to the side length of the pyramid:
l^2 = s^2/4 + s^2
l^2 = 5s^2/4
l = sqrt(5)s/2
The area of each triangular face is:
A = (1/2) * base * height
A = (1/2) * s * (sqrt(5)s/2)
A = (sqrt(5)/4)s^2
Since there are 4 triangular faces on the lateral surface of the pyramid, the total lateral surface area is:
L = 4*(sqrt(5)/4)s^2
L = sqrt(5)s^2
Since the skylight covering needs to match the lateral surface area of the pyramid, the amount of acrylic needed would be equal to the lateral surface area of the pyramid:
Acrylic needed = sqrt(5)s^2 square feet.