Finding cost of a triangular prism

Question

A hollow equilateral triangular prism has a fixed volume of 1.8m^3. The sides of the triangle have length P. To build the prism, one of the triangular faces costs 1.2 times what the remainder of the prism (excluding the base) costs, and the base costs 1.5 times what the remainder of the prism (excluding the previously mentioned triangular face) costs. How can the cost (C) of the prism be expressed in terms of P (the side length of the triangle)? What are the dimensions of the prism if the cost is minimum?

Answer 1

The triangle base is an equilateral triangle with side measure P.

Dropping the height, two congruent right triangles are formed

with angles 60 and 30. The height is sqrt(3)/4*P.

The volume is Base Area * Length

1.8 cubic meters = sqrt(3)/4 * P * Length

1.8*4 / (sqrt(3) * P) = length

7.2*sqrt(3)/ (3P) = length

2.4*sqrt(3)/ P = length

A side panel has dimensions P * length = 2.4*Sqrt(3)

Three (3) of these side panels are needed.

TWO (2) of the equilateral triangles are needed.

THe total surface area is

7.2*Sqrt(3) + sqrt(3)/2*P =

sqrt(3) * (7.2 + p/2)