Use the image to answer the question.
An illustration shows the net of a triangular prism. The illustration forms a shape where the small triangle is in the middle with each side being the base of three different triangles. The height of the triangle in the middle is 5 inches as shown by a dashed line and the base is 6 inches.
Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.2 .
(1 point)
To find the slant height of the triangular pyramid, first we need to calculate the lateral surface area of the triangular prism.
The lateral surface area of a triangular prism is given by the formula:
Lateral Surface Area = perimeter of base * height
Since the base of the prism is an equilateral triangle, the perimeter of the base is 3 times the length of one side.
Perimeter = 3 * 6 = 18 inches
Given that the height of the prism is 5 inches, we can calculate the lateral surface area:
Lateral Surface Area = 18 * 5 = 90 in^2
Since the surface area of the pyramid is 78 in^2, we need to subtract the area of the two triangular faces that form the top and bottom of the prism to find the lateral surface area of the triangular pyramid.
Each triangular face has an area of:
Area = 1/2 * base * height
Area = 1/2 * 6 * 5 = 15 in^2
There are two triangular faces, so the total area of these faces is 2 * 15 = 30 in^2
Subtracting this from the lateral surface area of the prism gives us the lateral surface area of the triangular pyramid:
90 - 30 = 60 in^2
The lateral surface area of a pyramid is given by the formula:
Lateral Surface Area = (perimeter of base * slant height) / 2
We know the lateral surface area is 60 in^2 and the perimeter of the base is 18 inches. Plugging this into the formula, we can solve for the slant height:
60 = (18 * slant height) / 2
60 = 9 * slant height
slant height = 60 / 9 = 6.67 inches
Therefore, the slant height of the triangular pyramid is 6.67 inches.