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Optimization Problems

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1 Answer

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]