Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.2 .

Answer 1

To find the slant height of the triangular pyramid, we need to use the formula for the surface area of a triangular pyramid:

Surface Area = 1/2 * perimeter of base * slant height + area of base

Since the pyramid is made of equilateral triangles, the base is an equilateral triangle. The formula for the area of an equilateral triangle is A = (sqrt(3) / 4) * s^2, where s is the side length of the triangle.

First, let's find the side length of the equilateral triangle base of the pyramid. Since the base is made of equilateral triangles, the perimeter of the base is 3 times the side length (3s). We can use the formula for the surface area of the pyramid to find the side length:

78 = 1/2 * 3s * slant height + (sqrt(3) / 4) * s^2

Simplify the equation:

78 = 3/2 * slant height * s + (sqrt(3) / 4) * s^2

Since we want to find the slant height, we will need to express s in terms of slant height using the equation for the surface area:

s = (78 - (sqrt(3) / 4) * s^2) / (3/2 * slant height)

Now we can substitute this expression for s into the equation for the surface area and solve for the slant height. This will give us the slant height of the triangular pyramid made of equilateral triangles.