Final answer:
The statement is true because the epiglottis is indeed made of elastic cartilage, which is vital for its function in closing the trachea during swallowing.
Step-by-step explanation:
The statement "The supporting structural material of the epiglottis is elastic cartilage" is true. The epiglottis is a leaf-shaped piece of elastic cartilage that is part of the larynx. This cartilage is very flexible and it plays a crucial role during swallowing by swinging to close the trachea, preventing food and beverages from entering the airway. The epiglottis, along with other cartilaginous structures like the thyroid and cricoid cartilages, form the major structure of the larynx, which also houses the true vocal cords and contributes to voice production.