The amount of space required by a particular firm is given below, where x and y

Question

The amount of space required by a particular firm is given below, where x and y are, respectively, the number of units of labor and capital utilized.
f(x,y) = 1000 sqrt(6x^2 + y^2)
Suppose that labor costs $480 per unit and capital costs $40 per unit and that the firm has $5000 to spend. Determine the amounts of labor and capital that should be utilized in order to minimize the amount of space required. (Hint: To minimize f(x, y) it is enough to minimize the function 6x2 + y2.)

Explanation / Answer

dsSVgrouch's approach is not wrong, but it doesn't use the Lagrange multiplier method - which I assume to be the point of your question. Also, he doesn't get to the answer. I'll show you how to use it: The function to be minimized is: f(x,y) = 1000 * sqrt(6x^2 + y^2) The constraint is: 5000 = 480x + 40y Define the constraining function: g(x,y) = 480x + 40y - 5000 The way to use the LM method is to look for a solution of the system of equations: 0 = g(x,y) 0 = ?*?g + ?f , where ? is an unknown constant. The first equation imposes the required constraint. The second equation says that the gradients of the contraint function and the optimized function are parallel to each other at the solution: This means that any small motion that is consistent with the constraint (and is thus perpendicular to the gradient of the constraint) is also perpendicular to the gradient of the optimized function - which means that the value of the optimized function is not changed by this motion. Since the value is not changed, it is a local extremum, consistent with the constraint, and is thus the answer being sought. OK, that's the theory. Now the calculation: Starting with: 0 = ?*?g + ?f x-equation: 0 = ?*?g/?x + ?f/?x .. = ?*(?/?x)(480x + 40y - 5000) + (?/?x)(1000 * sqrt(6x^2 + y^2)) .. = ?*480 + (2*x*6/2)(1000/sqrt(6x^2 + y^2)) .. = ?*480 + 6000x/sqrt(6x^2 + y^2) and y-equation: 0 = ?*?g/?y + ?f/?y .. = ?*(?/?y)(480x + 40y - 5000) + (?/?y)(1000 * sqrt(6x^2 + y^2)) .. = ?*40 + (2*y/2)(1000/sqrt(6x^2 + y^2)) .. = ?*40 + 1000y/sqrt(6x^2 + y^2) The x-equation leads to: ? = -(6000x/480)/sqrt(6x^2 + y^2) .. = -(25/2)x/sqrt(6x^2 + y^2) The y-equation leads to: ? = -(1000y/40)/sqrt(6x^2 + y^2) .. = -25y/sqrt(6x^2 + y^2) Together, they give: -(25/2)x/sqrt(6x^2 + y^2) = ? = -25y/sqrt(6x^2 + y^2) x/2 = y Now we go back to the constraint, knowing that: x = 2y 0 = g(x,y) = 480x + 40y - 5000 ............... = 480*2y + 40y - 5000 ............... = 1000y - 5000 => y = 5000/1000 = 5 => x = 2*5 = 10 That's the answer. Just for kicks, let's do it daSVgrouch's way: 5000 = 480x + 40y => y = (5000 - 480x)/40 = 125 - 12x f(x,y) = f(x, (125-12x) = 1000 *sqrt(6x^2 + (125 - 12x)^2) ........ = 1000 * sqrt(125^2 - 24*125x +144x^2 + 6x^2) ........ = 1000 * sqrt(125^2 - 3000x + 150x^2) We can maximize this with respect to x by just maximizing the interior of the sqrt: 0 = d ()/dx = -3000 + 300 x => x = 10 => y = 125 - 120 = 5 The answer is the same (of course, it has to be). But the Lagrange multiplier is more generally useful, because you often run into situations in which the constraint cannot easily be re-expressed as y = h(x) In such situations, you can still use the LM approach. (Of course, the problem could STILL get intractable, but it's another tool in the box!)

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