Final answer:
Rapid and shallow breathing at a rate of 40 breaths per minute in an elderly woman with rib injury poses a life threat due to the risk of inadequate oxygenation and hypoxemia, leading to potential organ damage, respiratory failure and complications like pneumonia.
Step-by-step explanation:
An elderly woman with a rapid and shallow breathing rate of 40 breaths per minute is likely experiencing a significant life threat. Rapid, shallow breathing can indicate a severe underlying issue such as a rib fracture, which can result in inadequate ventilation and subsequent hypoxemia (low levels of oxygen in the blood). This condition can lead to respiratory failure if not addressed promptly. In addition, inadequate oxygenation can cause vital organs, especially the brain, to become deprived of oxygen, leading to potential irreversible damage or death if prolonged. The physiological strain or pain from the injury along with difficulty drawing a full breath can exacerbate the situation, further reducing oxygen supply to the body. Consequently, this requires immediate medical attention to stabilize her breathing and prevent potential complications such as pneumonia, which would worsen her respiratory function.