Lagrange Multipliers and Constrained Optimization
Example: A rectangular box with an open top is to be designed for a volume of
700 in3. The material for its bottom cost 7¢ /in2 and the material for its four vertical
sides costs 5¢ /in2. Find the dimensions of the box that will minimize the cost of
the materials used in constructing the box.
Strategy: The first step is to determine the expression for the function and the
expression for the constraint. In this case the function to be minimized is the cost
function. Let the box have dimensions a, b, and c with the dimensions of the bottom
of the box being a by b. The dimension of the two sides are then a by c and
b by c.
Strategy: Determine the cost function, F, and the constraint function C.
F(a,b,c) = 7 a b + 2(5) a c + 2(5) b c = 7 a b + 10 a c + 10 b c
The constraint function is C(a, b, c). In this example the box is to be designed so
that the volume equals 700 in3 . So the constraint function is:
C(a, b, c) = a b c - 700 = 0
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