2022 AMC 8 Problem 9

A cup of boiling water (212^{\circ}\text{F}) is placed to cool in a room whose temperature remains constant at 68^{\circ}\text{F}. Suppose the difference between the water temperature and the room temperature is halved every 5 minutes. What is the water temperature, in degrees Fahrenheit, after 15 minutes?

\textbf{(A)} ~77\qquad\textbf{(B)} ~86\qquad\textbf{(C)} ~92\qquad\textbf{(D)} ~98\qquad\textbf{(E)} ~104

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