Question

"What constraints are placed on the variables in this situation, where a banquet hall is being set up for a reception? Each round table will have 6 chairs, and each rectangular table will have 10 chairs. The arrangement of tables must accommodate at least 185 guests to be seated. The client reserving the banquet hall requests that there be no more than 15 rectangular tables and no fewer than 6 round tables. Additionally, the number of rectangular tables used must be at least one-third the number of round tables used. Let x represent the number of round tables, and y represent the number of rectangular tables. Which of the following inequalities represents the constraints on these variables?"

Select the correct inequality from the options provided.

Answer 1

We use inequalities to express the constraints on the number of round and rectangular tables in a banquet hall, based on seating capacity and client requests. These constraints include seating at least 185 guests, using no more than 15 rectangular tables, no fewer than 6 round tables, and having rectangular tables be at least one-third the number of round tables.

Step-by-step explanation:

In mathematics, when dealing with constraints for real-world problems, we use inequalities to represent the limitations imposed on variables. For the situation described, where x represents the number of round tables, and y represents the number of rectangular tables, we have several constraints dictated by the banquet hall scenario:

• The total seating capacity must accommodate at least 185 guests, which translates to the inequality: 6x + 10y ≥ 185.
• The client requests no more than 15 rectangular tables: y ≤ 15.
• The client also requests no fewer than 6 round tables: x ≥ 6.
• The number of rectangular tables must be at least one-third the number of round tables used: y ≥ x/3.

These inequalities together form a system that defines the possible arrangements of tables within the given constraints. Solving this system will provide a combination of round and rectangular tables that meet the client's requirements and seat at least 185 guests.