Let x=long side, y=short side, z=height
(note: since the cost of material for the short side is more expensive, the minimization will automatically result in y being the shorter side without introducing an additional constraint of x-y>0, but do check when you get your answers)
The cost function would be:
subject to xyz=50.
I get $23,731.91 as the cost.
For your checking purposes, I get 12.64 inches as the sum of x,y and z.
Other Lagrange multiplier examples are available at this site at:
I think MathMate did not catch the fact that the box has to be 4 inches high, thus eliminating the variable z.
Cost = 300(8x+xy + 550(8y) + .02xy
= 2400x + 300xy + 4400y + .02xy
but V = 4xy
4xy = 50
y = 50/(4x)
cost = 2400x + 300x(50/4x) + 4400(50/(4x) + .02x(50/(4x))
= 2400x + 3750 + 55000/x + .25
d(cost)/dx = 2400 - 55000/x^2 = 0 for min of cost
2400 = 55000/x^2
x^2 = 55000/2400 = 22.916666
x = 4.79
y = 50/(4(4.79)) = 2.61
the box should be 4.79 in long, 2.61 in wide, and 4 inches high
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