A manufacturer produces a standard model and a deluxe model of a 13-in. television set. The standard requires 20 h of labor to produce, whereas the deluxe model requires 25 h. The labor available is limited to 300 h per week. In addition, the plant capacity is limited to producing 42 sets per week. Draw a graph of the region representing the number of sets that can be produced, given these conditions.
Which best describes this region?
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.
The region that represents the number of sets that can be produced given the conditions described can be defined by the following constraints:
1. Labor constraint: The total labor hours required to produce the standard model and the deluxe model should not exceed the labor available per week. The equation representing this constraint is: 20S + 25D ≤ 300, where S represents the number of standard models and D represents the number of deluxe models.
2. Plant capacity constraint: The total number of sets produced should not exceed the plant capacity per week. The equation representing this constraint is: S + D ≤ 42.
To graph this region, you can plot the points (S, D) that satisfy both constraints. Shade the area below and to the left of the line formed by the points to represent the feasible region. The feasible region will be bounded to the right by the S-axis, above by the D-axis, and by the lines that represent the constraints.
The best way to describe this region is as a shaded area in the first quadrant of the graph, limited by the S-axis, D-axis, and the lines formed by the constraints.