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# Confused

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A bit confused here

Julius  Mar 19, 2018
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Volume  =  L * W * H  = 70

But L  = 2W

So we have that

2W * W  * H   =  70

2W^2 * H  =  70

W^2 * H  =  35.......solve for the height

H  =  35/W^2

The surface area  is  the top  and bottom  = [ 2 * L * W]  = [ 2 *2W*W]  =  4W^2

And the total cost per m^2 for this part of the box is \$9 * Area  = 9*4W^2  =  36W^2

The surface area of two of the sides  =  2*2W * H  = 2*2W* (35/ W^2) =

140/W

And the surface area for the other two sides = 2*W*H = 2*W (35/W^2)  = 70/W

So....the total surface area of the sides  =  (140/W + 70/W)  =  210/W

And the total  cost per m^2  of this part of the box is \$6 * Area = 6*210/W  = 1260/W = 1260W^(-1)

So....the total cost, C,  is  given by

C  =  36W^2  + 1260W^(-1)      take the derivative and set to 0

C'  =  72W  - 1260W^(-2)  = 0

C'  = 72W - 1260/W^2  = 0

72W   =  1260/W^2

72W^3  =  1260

W^3  =  17.5

W  = 3√17.5  m

L  = 2W  =   2 3√17.5   m

H  =  35 /W^2  =   35 / 3√(17.5)^2   =  35 / 3√306.25   m

The minimum cost  is

36 W^2   +    1260 / W  =

36 3√[17.5^2]   +  1260 / 3√17.5   =

36 3√306.25  + 1260 / 3√17.5   ≈   \$727.97

CPhill  Mar 19, 2018
edited by CPhill  Mar 19, 2018

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