Problems which appear to be multivariable but include equality constraints may sometimes be reduced to single variable form.
Find the dimension which minimises total area.
Let the overall dimensions of the gauze be x cm by y cm. The objective function to be minimised is then clearly:
P = x y
The mounting margins provide the active width of (x-8) and height of (y-12), which must give a total exposed area of 300cm¼ so that there will be an equality constraint:
(x-8) (y-12) = 300
By rearrangement of the equality constraint we can eliminate either of the variable in the o.f. E.g. rearranging to get x:
x = (204+8y) / (4-12y)
The unconstrained single variable o.f. then is:
P = x y = y (204+8y) / (4-12y)
The minimum can be found either by differentiating and setting equal to zero or by a direct search.
W/V = (P1/K) [(P2/P1)K - 2 + (P3/P2)K]
Here K=(k-1)/k where k is the ration of heat capacities, approximately 1.4.
For the case where P1 = 1 atm and P3 = 4 atm confirm the well known result that P2 should be 2 atm.
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