Optimization Problems

Question

An open-top box is to be constructed so that its base is twice as long as it is wide. Its volume is to be 2300 cm^3. FInd the dimensions that will minimize the amount of cardboard required. 

 

Make:

Length = 2w

Width = w

Height = h

 

Show all your work. 

Answer 1

Width = W

Length = 2W
Height = h

 

Volume = 2300 cm³ = L*W*h = 2W*W*h = 2W²h

 

So h = 2300/2W² = 1150/W²

 

Area = 2*(W*h) + 2*(2W*h) + W*2W = 6900/W + 2W²        [Substituted 1150/W² for h, L = 2W]

 

To find the minimum Area, take the derivative of the AREA wrt W, set it to zero, and solve for W:

 

d(Area)/dW = -6900W^-2 + 4W

 

0 = -6900W^-2 + 4W

 

6900/4 = W³

 

1725 = W³

 

12 ≅ W                                               [11.993 rounded to 12]

 

L = 2W = 24                                      [23.986 rounded to 24]

 

h = 1150/W² = 8                              [7.995 rounded to 8]

 

CHECK:

 

Volume = 12*24*8 = 2304                 [Within rounding error]