Solve the problem. The surface area of a square pyramid is 116 in.2 and the total area of the pyramid’s four triangular faces is 80 in.2
What is the length of one of the sides?
I apologize for my previous response, I made an error. Here's a corrected explanation:
We have been given that the surface area of a square pyramid is 116 in.2 and the total area of the pyramid’s four triangular faces is 80 in.2. We need to find the length of one of the sides of the pyramid.
Let's start by using the formula for the total surface area of a square pyramid:
Total surface area = Area of the base + Sum of the areas of the four triangular faces
We know that the area of each triangular face is 20 in.2 (as given), so the sum of the areas of all four faces is 4*20 = 80 in.2.
We also know that the base of the pyramid is a square, so its area is s^2, where s is the length of one of its sides.
Therefore, we can write the total surface area of the pyramid as:
116 = s^2 + 80
Simplifying this equation, we get:
s^2 = 116 - 80 = 36
Taking the square root of both sides, we get:
s = 6 (the positive square root of 36)
So the length of one of the sides of the base of the pyramid is 6 inches.
Let's call the length of one of the sides "s".
We know the surface area of the pyramid is 116 in.2, which consists of the area of the square base and the four triangles.
The area of the square base is s^2.
The area of the four triangular faces is 80 in.2, so each triangle has an area of 20 in.2.
The base of each triangle is s (because the triangle connects to the square base), and we can find the height of each triangle by using the Pythagorean Theorem:
h^2 = s^2 - (s/2)^2
h^2 = s^2 - (s^2/4)
h^2 = (3s^2)/4
h = sqrt((3s^2)/4)
h = (sqrt(3)/2)s
Now we can find the surface area:
116 = s^2 + 4(1/2)(s)(sqrt(3)/2)s
116 = s^2 + 2s^2(sqrt(3))
116 = s^2(1 + 2sqrt(3))
s^2 = 116/(1 + 2sqrt(3))
s = sqrt(116/(1 + 2sqrt(3)))
s ≈ 4.77
So the length of one of the sides is approximately 4.77 inches.
i dont get it could u explain in a different way
Sure, I'll try to explain it in a different way:
We have been given that the surface area of a square pyramid is 116 in.2 and the total area of the pyramid’s four triangular faces is 80 in.2. We need to find the length of one of the sides of the pyramid.
Let's start by using the formula for the total surface area of a square pyramid:
Total surface area = Area of the base + Sum of the areas of the four triangular faces
We know that the area of each triangular face is 20 in.2 (as given), so the sum of the areas of all four faces is 4*20 = 80 in.2.
We also know that the base of the pyramid is a square, so its area is s^2, where s is the length of one of its sides.
Therefore, we can write the total surface area of the pyramid as:
116 = s^2 + 80
Simplifying this equation, we get:
s^2 = 116 - 80 = 36
Taking the square root of both sides, we get:
s = 6
So the length of one of the sides of the base of the pyramid is 6 inches.