To formulate the problem, let \( x \) represent the number of Standard chairs produced and \( y \) the number of Deluxe chairs produced.
Step 1: Formulating the inequalities
-
Construction and Finishing Time:
- Each Standard chair requires 2 hours.
- Each Deluxe chair requires 4 hours.
- The total time available for construction and finishing is 240 hours.
- Thus, the inequality is: \[ 2x + 4y \leq 240 \]
-
Upholstering Time:
- Each Standard chair requires 2 hours for upholstering.
- Each Deluxe chair requires 16 hours for upholstering.
- The total time available for upholstering is 480 hours.
- The inequality for upholstering is: \[ 2x + 16y \leq 480 \]
-
Non-negativity Constraints:
- The number of chairs produced cannot be negative: \[ x \geq 0, \quad y \geq 0 \]
Summary of Inequalities:
The system of inequalities is: \[ \begin{align*}
- & \quad 2x + 4y \leq 240 \
- & \quad 2x + 16y \leq 480 \
- & \quad x \geq 0 \
- & \quad y \geq 0 \ \end{align*} \]
Part A: Identifying the Correct Option
The correct answer would be the formulation above, which corresponds to option D:
- Left brace Start 3 By 1 Matrix 1st Row 1st Column \( 2 x + 4 y \leq 240 \) 2nd Row 1st Column \( 2 x + 16 y \leq 480 \) 3rd Row 1st Column \( x \geq 0, , y \geq 0 \) End Matrix
Step 2: Graphing the Inequalities
To identify the solution region graphically, we would:
- Graph the lines represented by the equations \( 2x + 4y = 240 \) and \( 2x + 16y = 480 \).
- Identify the area that satisfies both inequalities, as well as the non-negativity constraints.
Finding Points of Intersection
- Rearranging the equations to find intercepts would be helpful.
-
From \( 2x + 4y = 240 \):
- Setting \( x = 0 \), \( y = 60 \) (intercept)
- Setting \( y = 0 \), \( x = 120 \) (intercept)
-
From \( 2x + 16y = 480 \):
- Setting \( x = 0 \), \( y = 30 \) (intercept)
- Setting \( y = 0 \), \( x = 240 \) (intercept)
Identify vertices for the feasible region:
To find the corners, solve the system of equations at intersections at \( (0, 0) \), \( (0, 30) \), \( (0, 60) \) and by finding the intersection of the two lines themselves:
Set \( 2x + 4y = 240 \) equal to \( 2x + 16y = 480 \):
- From \( 2x + 4y = 240 \), we can rearrange to find \( y \): \[ y = 60 - \frac{x}{2} \]
Substituting into \( 2x + 16(60 - \frac{x}{2}) = 480 \): \[ 2x + 960 - 8x = 480 \Rightarrow -6x = -480 \Rightarrow x = 80 \Rightarrow y = 30 \] Thus, you have points \( (0, 0) \), \( (0, 30) \), \( (120, 0) \), and \( (80, 30) \), which can be plotted.
Part B: Identify corners of the region
The corners of the feasible region would be:
- \( (0, 0) \)
- \( (0, 30) \)
- \( (120, 0) \)
- \( (80, 30) \)
These intersections will shape your feasible region, typically forming a polygon in the first quadrant. The graph can be constructed with the constraints described above.
1 answer