An open rectangular box having a volume of 4 in.3 is to be constructed from cardboard. find the dimensions of such a box if the amount of cardboard used in its construction is to be minimized

8.1 Function of Two Variables Many functions have several variables. Ex. There are 3 types of football tickets. Type A costs $50, type B costs $30, and type C costs $20. If in a match, x tickets of type A, y tickets of type B, and z tickets of type C are sold, the total income of ticket sale is f(x, y, z) = 50x + 30y + 20z. Def. A real-valued function of two variables f consists of 1. A domain A consisting of ordered pairs of some real numbers (x, y). 2. A rule that associates with each ordered pair in A with one real number, denoted by z = f(x, y). Ex. (Ex 1, p.532) f(x, y) = x + xy + y w + 2. Compute f(0, 0), f(1, 2), and f(2, 1). Ex. Find the domain of a. f(x, y) = x 2+y 2 , b. g(x, y) = 2 x−y , c. h(x, y) = p 1 − x 2 − y 2. Ex. Find the domain of g(r, s) = √ rs. The graph of a function z = f(x, y) is the collection of all points {(x, y, f(x, y)) : x, y ∈ A} in R 3 (Fig 5, p.534; Fig 6, p.535). The graph of z = f(x, y) is 3 dimensional and it is difficult to draw. So we use level curves. A level curve is the graph of c = f(x, y) on xy-plane for a constant c. By drawing the level curves corresponding to several admissible values of c, we obtain a contour map. (Fig 7, p.535; Fig 8, p.536) Ex. Ex 5, p.536. HW. C8.1: SC1, 2, 3, Ex 31, 33