Graph of functions with two variables (continued)

 In this post, we continue with examples. We are going to solve a practical problem concerning nuts and bolts.

Problem

A profit function for a hardware manufacturer is given by:

where x is the number of nuts sold per month (measured in thousands) and y the number of bolts sold per month (measured in thousands). Profit is measured in thousands of dollars. Sketch a graph of this function

Solution

Let's determine the domain of the function. This function is a polynomial function with two variables. For a profit to occur, we need to have f(x,y) ≥ 0. In other terms:

 

 

This is a disk of radius 4 centered at the point (3, 2) where x and y must be non-negative. When x = 3 and y = 2 f(x, y) = 16. Note that x and y can be non-integers. For example it is possible to sell 2.5 thousands nuts per month. The domain contains thousands of points. We can consider all points within the disk. For any z<16, we can solve the equation f(x,y) = z.

The second side of the last previous equation represents the square of the radius of the circle  with center (3,2). Therefore this expression must be strictly superior to zero.: 16-z>0. Therefore z<16. 

The range of f(x,y) is:

The graph of f(x,y) is a paraboloid pointed downward.