An open box is to be constructed so that the length of the base is 4 times large

Question

An open box is to be constructed so that the length of the base is 4 times larger than the width of the base. If the cost to construct the base is 2 dollars per square foot and the cost to construct the four sides is 1 dollars per square foot, determine the dimensions for a box to have volume = 70 cubic feet which would minimize the cost of construction. The values for the dimension of the base are: ?, ? The height of the box is: ?

Explanation / Answer

total cost = cost of base + cost of sides = (w*5*w + 2*w*h + 2*5w*h)*3 c = (5ww+12wh)*3 since the volume is 25 ft cube, w*5w*h=25===> wh= 5/w therefore c= (15w^2 + 180/w ) dc/dw = 30w -180/w^2 for optimized width 0= 30w - 180/w^2 w = cube root of 60 l= 5* cube root of 60 h= 25/ 5* cube root of 60* cube root of 60 = 5/ 60 ^ (2/3)

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