Final answer:
The correct route for taking the temperature of a 2-year-old child with diarrhea and an ear infection is the tympanic method, using an infrared ear thermometer. Option D is the correct answer.
Step-by-step explanation:
The most appropriate route for obtaining the temperature of a 2-year-old child with diarrhea and a right ear infection would be the tympanic method. This method makes use of an infrared ear thermometer developed by Diatek Corporation with support from NASA. It leverages infrared astronomy technology to measure the temperature emanating from the eardrum, similar to how astronomers measure the temperature of stars and planets. This method is highly suitable as it is quick, reduces the risk of cross infection by avoiding contact with mucous membranes, and can be used in patients that are newborn, critically-ill, or incapacitated.