Question

The Baraboo, Sheffield, plant of International Widget Ltd. uses aluminium, iron and magnesium to produce high-quality widgets. The quantity of widgets that may be produced using x tons of aluminium, y tons of iron and z tons of magnesium is Q(x, y, z) = x yz. The cost of raw materials is aluminium 6 per ton; iron 4 per ton; and magnesium 8 per ton. How many tons each of aluminium, iron and magnesium should be used to manufacture 1000 widgets at the lowest possible price? Hint: You want an extreme value for what function? Subject to what constraint?

Answer 1

We are looking for minima of the cost function

f (x, y, z) = 6x + 4y + 8z

subject to the constraint Q(x, y, z) = 1000, where Q(x, y, z) = x yz. Using Lagrange multipliers the equations for critical points are

6 = yzλ

4 = xzλ

8 = xyλ

xyz = 1000

We multiply the first equation by x, the second by y and the third by z to obtain

We substitute this into the constraint equation to obtain

We substitute this into the constraint equation to obtain

Therefore the least cost for producing 1000 widgets is achieved using

tons of aluminium

tons of iron and

tons of magnesium