SOLUTION: a company wants to manufature gift boxes in the shape of rectangular prisms .each gift box will have a volume of 84 cubic inches the base of the rectangular prism shold be twice as
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Question 1121065: a company wants to manufature gift boxes in the shape of rectangular prisms .each gift box will have a volume of 84 cubic inches the base of the rectangular prism shold be twice as long as it is wide.what dimensions should the company choose for the gift boxes in order to minimize the surface area of each box.
Found 2 solutions by solver91311, htmentor:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Width of base: 
Length of base: 
Height of prism: 
Volume of prism: \ =\ 2w^2h\ =\ 84)
Surface Area: \ =\ 4w^2\ +\ 4wh\ +\ 2wh\ =\ 4w^2\ +\ 6wh)
But then: 
Hence the surface area as a function of the width is \ =\ 4w^2\ +\ \frac{252}{w})
And then the first derivative of the surface area function is:
\ =\ 8w\ -\ \frac{252}{w^2})
Set the derivative equal to zero:
Solve for , then calculate and 
John
My calculator said it, I believe it, that settles it
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
The volume of the rectangular prism is V = l*w*h = 84 in^3
The area is A = 2(l*w + l*h + w*h)
The base l is equal to twice the width: l = 2w
Thus V = 2*w^2*h and A = 2(w^2 + 2wh + wh) = 2w(w + 3h)
Expressing h in terms of w gives h = V/(2*w^2)
Thus A = 2w(w + 3V/(2*w^2))
The area will be minimized when dA/dw = 0
0 = 4w - 3V/w^2 -> w = (3V/4)^(1/3)
Substituting the value for V gives w = 3.979
Therefore l = 7.958 and h = 84/(2*3.979^2) = 2.653
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