AMC 12 2020b Problem 9

A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));

label("$4$", (2,0), S);

\textbf{(A)}\ 3\pi \sqrt5 \textbf{(B)}\ 4\pi \sqrt3 \textbf{(C)}\ 3 \pi \sqrt7 \textbf{(D)}\ 6\pi \sqrt3 \textbf{(E)}\ 6\pi \sqrt7