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The respiratory system consists largely of your lungs and functions to provide the gaseous exchange between oxygen and carbon dioxide. Animals have to have oxygen to live, they create carbon dioxide as a waste product. The circulatory system consists of your heart, all your blood vessels, and the blood that is contained within your body. The purpose of the circulatory system is to act as a "super highway", transferring in nutrients and oxygen to each and every cell in your body. Cellular respiration takes the nutrients (glucose) and combines it with oxygen to release energy in the form of ATP. Cellular respiration also creates carbon dioxide and water vapor as waste products.
So the function these two systems seem to have in common is they both provide necessary parts for assisting in cellular respiration. The respiratory system pulls in oxygen, while it takes out the "trash", carbon dioxide and water vapor. The circulatory system provides a "round trip" commute, providing a path for the oxygen to get in and the carbon dioxide to get out.
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