2017 AMC 12A Problem 6

Joy has 30 thin rods, one each of every integer length from 1 \text{ cm} through 30 \text{ cm}. She places the rods with lengths 3 \text{ cm}, 7 \text{ cm}, and 15 \text{cm} on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?

\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 20

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.