Question

Mr. Ollyin is buying new pillows for his hotel. He buys three types of pillows: soft, medium and firm. He must buy at least 100 soft pillows and at least 200 medium pillows. He must buy at least 400 pillorss in total. Soft pillows cost $4 each. Medium pillows cost $3 each. Firm pillows cost $4 each. He wishes to spend no more than $1800 on new pillows. At least 40% of the new pillows must be medium pillows. Ollyin buys x soft pillows, y medium pillows and z firm pillows. (a) In addition to x≥0,y≥0 and z≥0, find five inequalities in x,y and z that model the above constraints.

Answer 1

The constraints provided for purchasing pillows can be represented by the following inequalities: x ≥ 100, y ≥ 200, x + y + z ≥ 400, 4x + 3y + 4z ≤ 1800, and y ≥ 0.4(x + y + z) to indicate that at least 40% of the pillows must be medium.

Step-by-step explanation:

The question asks for a mathematical model using inequalities to represent the constraints for purchasing pillows for a hotel. According to the constraints, we can establish the following inequalities:

• x ≥ 100 (at least 100 soft pillows)
• y ≥ 200 (at least 200 medium pillows)
• x + y + z ≥ 400 (at least 400 pillows in total)
• 4x + 3y + 4z ≤ 1800 (spending no more than $1800)
• Since at least 40% must be medium pillows: y ≥ 0.4(x + y + z)

These inequalities must all be satisfied simultaneously in order for Mr. Ollyin to meet the purchasing requirements for his hotel pillows while also adhering to his budget.