I would say that this problem is missing some essential information, namely, an explicit link between cost and surface area.
For example, there is a solution if you assume that 'cost' means 'cost per square meter of surface area,' so I will proceed, given that assumption.
Let's work out the surface area geometry first. Here is what we know:
- The prism volume is 1.8 cubic meters
- The ends of the prism are equilateral triangles, with sides of length P.
We need to calculate the area of the triangles, and the length of the prism.
Use Pythagoras to calculate the height of the triangular ends --- you get h = (P*√3) / 2.
Then the area of each triangular end, TA, is 1/2 times base times height --- TA = (P2 * √3) / 4.
Let L be the Length of the prism. We know the volume, V, of the prism, so we'll use that to calculate the length.
The Volume of a prism is just the area of the ends times the length of the prism.
V = TA * L
Rearranging, we have
L = V / TA
Plugging in the values for V and TA, we get
L = 1.8 / ((P2 * √3) / 4)
L = (4 * 1.8) / (P2 * √3)
Then the area of one of the 3 side facets is just P * L:
SF = (4 * 1.8) / (P * √3)
Now for the Cost. We're not given the cost directly, but let K be the cost per square meter of the 3 lengthwise Side Facets. Then the Total Cost (TC) for the 3 side facets plus the 2 triangular ends is given by:
TC = (K * 3 * SF) + (K * 1.2 * TA) + (K * 1.5 * TA)
After substituting, combining and simplifying, I get
TC = K * [(7.2 * √3) / P + (2.7 * √3 * P2) / 4]
NOW, we're asked to minimize the total cost function:
f(P) = [(7.2 * √3) / P + (2.7 * √3 * P2) / 4]
(We don't need the K, since it's just a constant and doesn't affect the location of the minimum.)
But our function is not a parabola, so we don't have any immediate way to solve for P. Instead, I used a graphing calculator to calculate that the cost will be a minimum when P = 1.747 meters. (Unless I made a mistake somewhere, which is quite possible!)