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Zorluk: OrtaSystems of Linear Inequalities in Two Variables

A manufacturer produces standard chairs and deluxe chairs. Each standard chair requires 22 hours of assembly and 11 hour of finishing. Each deluxe chair requires 33 hours of assembly and 22 hours of finishing. The manufacturer has a maximum of 240240 hours available for assembly and a maximum of 150150 hours available for finishing each day. Additionally, the manufacturer must produce at least 1010 deluxe chairs daily. If xx represents the number of standard chairs produced daily and yy represents the number of deluxe chairs produced daily, which of the following systems of inequalities best represents this situation?

  1. A
    2x+3y≥240x+2y≥150y≥10\begin{aligned} 2x + 3y &\geq 240 \\ x + 2y &\geq 150 \\ y &\geq 10 \end{aligned}
  2. B
    3x+2y≤2402x+y≤150y≥10\begin{aligned} 3x + 2y &\leq 240 \\ 2x + y &\leq 150 \\ y &\geq 10 \end{aligned}
  3. 2x+3y≤240x+2y≤150y≥10\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\geq 10 \end{aligned}Cevap
  4. D
    2x+3y≤240x+2y≤150y≤10\begin{aligned} 2x + 3y &\leq 240 \\ x + 2y &\leq 150 \\ y &\leq 10 \end{aligned}

Cevap

The system containing 2x+3y≤2402x + 3y \leq 240, x+2y≤150x + 2y \leq 150, and y≥10y \geq 10.
The correct system represents each constraint accurately. The assembly time requirement is 2x+3y≤2402x + 3y \leq 240 because standard chairs consume 22 hours each, deluxe chairs consume 33 hours each, and the total cannot exceed 240240. The finishing time requirement is x+2y≤150x + 2y \leq 150 since standard chairs consume 11 hour each, deluxe chairs consume 22 hours each, and the total cannot exceed 150150. Finally, the requirement to produce at least 1010 deluxe chairs means y≥10y \geq 10.

Adım Adım Çözüm

1
Set up the inequality for assembly time.
2x+3y≤2402x + 3y \leq 240
Each standard chair (xx) needs 22 hours and each deluxe chair (yy) needs 33 hours, with a maximum limit of 240240 hours.
2
Set up the inequality for finishing time.
x+2y≤150x + 2y \leq 150
Each standard chair (xx) needs 11 hour and each deluxe chair (yy) needs 22 hours, with a maximum limit of 150150 hours.
3
Set up the inequality for the minimum production limit of deluxe chairs.
y≥10y \geq 10
The manufacturer must produce at least 1010 deluxe chairs, meaning the quantity must be greater than or equal to 1010.
4
Combine the individual inequalities into a single system.
The final system is 2x+3y≤2402x + 3y \leq 240, x+2y≤150x + 2y \leq 150, and y≥10y \geq 10.
All three conditions must be satisfied simultaneously.

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Systems of Linear Inequalities in Two Variables
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Systems of Linear Inequalities in Two Variables

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