To draw a graph representing the region of possible production quantities, we need to plot the constraints given in the problem.
Let's consider the standard model as x and the deluxe model as y. We have the following constraints:
1. Labor constraint: The standard model requires 20 h of labor, and the deluxe model requires 25 h. The total labor available per week is limited to 300 h. So the equation for the labor constraint is: 20x + 25y ≤ 300.
2. Plant capacity constraint: The plant can produce a maximum of 42 sets per week. So the equation for the plant capacity constraint is: x + y ≤ 42.
To graph these constraints, we need to plot the lines for each constraint and shade the region that satisfies both constraints.
Start by plotting the line 20x + 25y = 300. To do this, rearrange the equation to y = (300 - 20x) / 25. Pick some x-values (let's say 0, 10, and 20) and substitute them into the equation to find the corresponding y-values. Plot the points and draw a line through them.
Next, plot the line x + y = 42. Rearranging the equation to y = 42 - x, pick some x-values (again, let's say 0, 10, and 20) and substitute them into the equation to find the corresponding y-values. Plot the points and draw a line through them.
Now, shade the region that satisfies both constraints. This region represents the combination of x and y values that satisfy the labor and plant capacity limits.
The best way to visualize this region is by shading it on a graph, but as an AI text-based bot, I cannot directly show you a graph. However, I hope you understand the concept and the process of graphing the constraints to find the region of possible production quantities.