9. Optimization Problem Suppose an airline policy states that all baggage must b
ID: 2891673 • Letter: 9
Question
9. Optimization Problem Suppose an airline policy states that all baggage must be box shaped with a sum of length, width, and height not exceeding 120 inch. What are the dimensions and volume of a square- based box with greatest volume under these conditions? (a) Identify the objective function. (b) Identify the constraint (s). (c) Express the objective function in terms of one variable using the constraint(s). (d) Find the interval of interest of the objective function (e) Find the dimensions of the container with greatest volume. Maximum volume: V = Dimensions: w .Explanation / Answer
Solution:
l = w
l + w + h < 120
Let's assume that l + w + h = 120 will give us the greatest possible volume, just for the sake of simplicity
V = l * w * h
V = w * w * h
V = w2 * h
l + w + h = 120
w + w + h = 120
h = 120 - 2w
V = w2 * (120 - 2w)
V = 2 * w2 * (60 - w)
V = 2 * (60w2 - w3)
dV/dw = 2 * (120w - 3w2)
dV/dw = 2 * 3 * (40w - w2)
dV/dw = 0
0 = 6 * w * (40 - w)
0 = w * (40 - w)
w = 0 , 40
l = w = 40
l + w + h = 120
40 + 40 + h = 120
h = 40
40 by 40 by 40 will give you the largest box (that's a pretty big box, too, as it's 64 cubic feet, which is about 3 times the size of a normal refrigerator's capacity)
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