A pastry chef is creating a chocolate dessert in the shape of a triangular pyramid. The pyramid has a base of 5 inches and the height of the base is 4.3 inches and the slant height is 6 inches use the correct formula to find the surface area of the chocolate pyramid.
To find the surface area of a triangular pyramid, we need to find the area of the base and the area of the four triangular faces.
The formula to find the area of a triangle is:
Area = (1/2) * base * height
First, let's find the area of the base:
Base = 5 inches
Height of the base = 4.3 inches
Area_base = (1/2) * 5 inches * 4.3 inches = 10.75 square inches
Next, let's find the area of the four triangular faces.
The slant height is given as 6 inches.
Area of each triangular face = (1/2) * base * height
Since the base and height of each triangular face are not given, we need to find them using the given measurements.
The slant height of a triangular face can be calculated using the Pythagorean theorem:
slant height = sqrt( (base/2)^2 + height^2 )
6 inches = sqrt( (base/2)^2 + 4.3 inches^2 )
Simplifying the equation, we get:
36 inches^2 = (base/2)^2 + 18.49 inches^2
Subtracting 18.49 inches^2 from both sides, we get:
17.51 inches^2 = (base/2)^2
Taking the square root on both sides, we get:
4.18 inches = base/2
Multiply both sides by 2, we get:
8.36 inches = base
Now, we can calculate the area of each triangular face:
Area_triangular_face = (1/2) * base * height
Area_triangular_face = (1/2) * 8.36 inches * 4.3 inches = 18.0 square inches
Since there are four triangular faces, the total area of the four triangular faces is:
Total_area_triangular_faces = 4 * Area_triangular_face = 4 * 18.0 square inches = 72.0 square inches
Finally, we can find the surface area of the chocolate pyramid by summing the area of the base and the areas of the four triangular faces:
Surface_area = Area_base + Total_area_triangular_faces
Surface_area = 10.75 square inches + 72.0 square inches = 82.75 square inches
Therefore, the surface area of the chocolate pyramid is 82.75 square inches.