Use the web to find the inclined height of the triangular pyramid made of equilateral triangles if its surface area is 78 inches^2. There is a triangle network with 4 triangles with a dotted line in the middle of the triangle in the center with a height of 5 and a base of 6 in the central triangle. The formula is:
SA = (1/2 base x height) + (1/2 perimeter x inclined height)
First, we need to find the base and height of the equilateral triangle. The formula for the area of an equilateral triangle is A = (sqrt(3) / 4) * s^2, where s is the side length.
Given that the central triangle has a base of 6 inches, we can use this information to find the side length of the equilateral triangle. Since the center triangle is equilateral, all sides will have the same length, which will be 6 inches.
So, the area of the equilateral triangle is A = (sqrt(3) / 4) * 6^2 = (sqrt(3) / 4) * 36 = 9sqrt(3) square inches.
Since there are 4 equilateral triangles making up the pyramid, the surface area is 4 times the area of one equilateral triangle, which is 4 * 9sqrt(3) = 36sqrt(3) square inches.
Given that the surface area is 78 square inches, we can set up the equation:
78 = (1/2 * 6 * 5) + (1/2 * 18 * h)
Solving for h, the inclined height of the pyramid:
78 = 15 + 9h
63 = 9h
h = 7
Therefore, the inclined height of the triangular pyramid is 7 inches.