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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?

Respuesta :

trccater54

   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

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