GMAT Linear Inequalities practice questions

A bicycle manufacturer produces both standard and deluxe models. The production facility can assemble at most 40 bicycles per day. The standard model requires 2 hours of labor, while the deluxe model requires 3 hours. The facility has 90 labor-hours available each day. If the profit per standard model is $80 and per deluxe model is $120, how many deluxe models should be produced daily to maximize profit while not exceeding any constraints? Additionally, the manufacturer must produce at least 5 standard models each day to fulfill existing orders.

A hospital's emergency department has two types of patient arrivals: critical cases arriving at a rate of 8 per hour and non-critical cases arriving at 15 per hour. The department can handle up to 20 patients per hour. To manage overflow, the hospital will divert non-critical cases to a nearby clinic if the total expected arrivals exceed capacity. If the current hour has already seen 5 critical cases arrive, what is the maximum number of additional non-critical cases that can be accepted in the next 30 minutes without triggering diversion?

A hospital must ensure its emergency department has enough ventilators to handle patient surges. The current inventory is 45 ventilators. Based on patient volume projections, the hospital determines that for every 10% increase in patient volume beyond current capacity, 3 more ventilators are needed. If the hospital needs to prepare for a maximum patient volume increase of p%, where p is a positive integer, what is the largest value of p such that the ventilator requirement does not exceed twice the current inventory?

A hospital is planning its staffing for the next quarter. The hospital currently has 120 nurses and needs to ensure that the number of nurses on duty each day does not exceed 90% of its total nursing staff. Additionally, the hospital wants to maintain at least 60 nurses on duty each day. If the hospital hires additional nurses, what is the minimum number of additional nurses required to meet these constraints?

A farmer is planning the optimal planting strategy for two crops, X and Y, based on irrigation constraints. Each acre of crop X requires 3 units of water per week, while each acre of crop Y requires 5 units. The farm's irrigation system can deliver at most 120 units of water per week. To maintain soil health, the farmer must plant at least 5 acres of crop X. If the farmer aims to maximize the total planted area $X + Y$, what is the maximum number of acres that can be planted under these constraints?