What is the appropriate oxygen delivery method for a 45-year-old female experiencing difficulty breathing and cyanosis on the lips?

The appropriate oxygen delivery method in this scenario is to provide 15 liters per minute of oxygen via a non-rebreather mask. This choice is optimal for a patient who is experiencing difficulty breathing and cyanosis, which indicates a potential lack of adequate oxygenation.

A non-rebreather mask is designed to deliver high concentrations of oxygen—typically around 90% or more. It achieves this by providing a reservoir bag that fills with pure oxygen, allowing the patient to inhale this high-flow oxygen while minimizing the re-breathing of exhaled carbon dioxide. In cases of respiratory distress and cyanosis, it is crucial to rapidly and effectively increase the patient's oxygen saturation levels.

The other options would not provide sufficient oxygen flow or concentration for someone with significant respiratory distress. For example, lower flow rates provided by nasal cannulas, venturi masks, or simple masks may not adequately address the urgent need for high-flow oxygen in patients showing signs of severe hypoxia or respiratory failure. Thus, the non-rebreather mask is the most effective method to manage this patient's immediate respiratory needs.