Optimization
1. A rectangular box is resting on the xy-plane with one vertex at the origin. The opposite vertex
lies in the plane 6x + 4y + 3z = 24. Find the maximum volume of such a box.
2. An electronics manufacturer determines that the profit P (in dollars) obtained by producing
and selling x units of an LCD television and y units of a plasma television is approximated by the
model P(x, y) = 8x +10 y -(0.001)(x 2 + xy + y 2 ) -10,000 . Find the production level that produces a
maximum profit. What is the maximum profit?
3. Find the minimum distance from (0,0,0) to the plane x – y + z = 3. (Hint: minimize the square of
the distance.)
4. Find three positive integers x, y, and z whose product is 27 and sum is a minimum.
5. Find three positive integers x, y, and z whose sum is 30 and sum of the squares is a minimum.
6. A home improvement contractor is painting the walls and ceiling of a rectangular room. The
volume of the room is 668.25 cubic feet. The cost of wall paint is $0.06 per square foot and the
cost of ceiling paint is $0.11 per square foot. Find the room dimensions that result in a minimum
cost for the paint. What is the minimum cost for the paint?
7. A company manufactures running shoes and basketball shoes. The total revenue from x1 units
of running shoes and x2 units of basketball shoes is R = -5x12 -8x22 - 2x1x2 + 42x1 +102x2 where x1
and x2 are in thousands of units. Find x1and x2 so as to maximize the revenue.