The volume of the box must be 100 cubic meters. the cost of the material to be used on the sides is $11 per square meter and the cost of the material to be used on the bottom is $19 per square meter. what is the minimum cost?

   V = Lwh 
                  10 = (2w)(w)(h) 
                  10 = 2hw^2 
                    h = 5/w^2

Cost:    C(w) = 10(Lw) + 2[6(hw)] + 2[6(hL)]) 
                       = 10(2w^2) + 2(6(hw)) + 2(6(h)(2w) 
                       = 20w^2 + 2[6w(5/w^2)] + 2[12w(5/w^2)] 
                       = 20w^2 + 60/w + 120/w 
                       = 20 w^2 + 180w^(-1) 
             C'(w) = 40w - 180w^(-2)

Critical numbers: 
           (40w^3 - 180)/w^2 = 0 
                       40w^3 -180 = 0 
                               40w^3 = 180 
                                   w^3 = 9/2 
                                       w = 1.65 m 
                                        L = 3.30 m 
                                        h = 1.84 m

Cost:  C = 10(Lw) + 2[6(hw)] + 2[6(hL)]) 
               = 10(3.30)(1.65) + 2[6(1.84)(1.65)] + 2[6(1.84)(3.30)]) 
               = $165.75                       cheapest cost

ca = ec

cpctc makes it so that corresponding parts of the triangle you proved to be equal are equal, the other options aren't part of the part of the triangles you proved to be equal.