Answer:
1a) Vertex 1 = (0, 0)
1b) Vertex 2 = (80, 80)
1c) Vertex 3 = (132, 54)
1d) Vertex 4 = (80, 0)
2a) R = 0
2b) R = 336
2c) R = 378.9
2d) R = 156
3) 132 standard-mix packages and 54 deluxe-mix packages.
Explanation:
A confectioner sells two types of nut mixtures.
Let x be the number of standard-mix packages sold.
Let y be the number of deluxe-mix packages sold.
The revenue produced by selling the packages is given by the equation:

We have been given five constraints represented by this system of inequalities:

To maximize the revenue produced by selling the packages subject to the given constraints, we need to analyse the feasible region defined by the system of inequalities.
The feasible region for the given constraints is the region where all the shaded areas overlap. Therefore, we need to graph the system of inequalities given by the constraints.
Rearrange the first two inequalities so that they are in slope-intercept form:

When graphing an inequality with the sign ≥, draw a solid line and shade above the line.
When graphing an inequality with the sign ≤, draw a solid line and shade below the line.
Graph the inequalities (see attachment):
- Draw a solid line at y = -(1/2)x + 120 and shade below the line.
- Draw a solid line at y = -3x + 450 and shade below the line.
- Draw a solid line at y = x and shade below the line.
- For x ≥ 0 shade to the right of the y-axis.
- For y ≥ 0 shade above the x-axis.
The feasible region has four corner points (vertices).
Vertex 1 is the point of intersection of y = x, the x-axis and the y-axis.
Therefore, vertex 1 is the origin (0, 0).
Vertex 2 is the point of intersection between y = x and x + 2y = 240. Substitute the first equation into the second equation and solve for x. Then substitute the found value of x into the first equation and solve for y:

Therefore, vertex 2 is (80, 80).
Vertex 3 is the point of intersection between x + 2y = 240 and 3x + y = 450. Rearrange the second equation to isolate y, substitute it into the first equation and solve for x. Then substitute the found value of x into the second equation and solve for y:

Therefore, vertex 3 is (132, 54).
Vertex 4 is the point of intersection of 3x + y = 240 and the x-axis.
Substitute y = 0 into the equation and solve for x:

Therefore, vertex 4 is (80, 0).
In summary, the four vertices are:
- Vertex 1 = (0, 0)
- Vertex 2 = (80, 80)
- Vertex 3 = (132, 54)
- Vertex 4 = (80, 0)
Determine the value of R at each vertex by substituting the x and y values of the points into the equation for R:

The maximum value of R is $378.90 at vertex (132, 54).
Therefore, the confectioner needs to sell 132 standard-mix packages and 54 deluxe-mix packages to maximize her revenue.