The bases of a triangular prism are equilateral triangles. The surface area of the prism is 80 square units. If the area of one of the triangular bases is half the area of one of the rectangular faces, what is the area of one of the triangular bases of the prism?

base area = B

face area = 2B
total surface (two bases + 3 faces):

2B + 3(2B) = 80

To find the area of one of the triangular bases of the prism, we need to follow these steps:

Step 1: Recall that a triangular prism consists of three rectangular faces and two triangular bases.

Step 2: Let's assume that the area of one of the triangular bases is A square units. Since both bases are equilateral triangles, the total area of the two bases will be 2A square units.

Step 3: The problem states that the area of one of the bases is half the area of one of the rectangular faces. Let's call the area of one rectangular face F square units.

Step 4: Since there are three rectangular faces, the total area of the three rectangular faces will be 3F square units.

Step 5: The problem further states that the surface area of the prism is 80 square units. Therefore, we have the equation:

2A + 3F = 80

Step 6: We know that the area of an equilateral triangle with side length 's' is given by the formula A = (sqrt(3)/4) * s^2.

Step 7: Since the bases are equilateral triangles, the area of one triangular base can be expressed as A = (sqrt(3)/4) * s^2, where 's' is the side length of the triangle base.

Step 8: Now, let's solve the equation using the given information. Since the area of one of the bases is half the area of one of the rectangular faces, we can write A = (1/2)F.

Step 9: Substituting this into the previous equation, we get:

2(1/2)F + 3F = 80
F + 3F = 80
4F = 80
F = 20

Step 10: Now that we have found the area of one rectangular face (F = 20 square units), we can substitute this value back into the equation for the triangular base:

A = (1/2)F
A = (1/2)(20)
A = 10

Step 11: Therefore, the area of one of the triangular bases of the prism is 10 square units.