Find the area of the part of the plane 5x + 4y + z = 20 that lies in the first octant.

This part of the plane is a triangle. Call it . We can find the intercepts by setting two variables to 0 simultaneously; we'd find, for instance, that means , so that (4, 0, 0) is one vertex of the triangle. Similarly, we'd find that (0, 5, 0) and (0, 0, 20) are the other two vertices.

Next, we can parameterize the surface by



so that the surface element is



Then the area of is given by the surface integral

The answer would be 27 because assuming that each package has 1 bar of soap and 1 tube of toothpaste, after 27 packages they would run out of toothpaste