Solution:

Remember that the volume of a cone is equal to $\dfrac 13 \pi r^2 h.$ Further notice that the circumference of the base of the cone is equal to the remaining circumference of the circle: $\dfrac 34$ the circumference of the complete circle.

Therefore, the circumference of the base of the cone is equal to: $$\begin{align*} C &= \dfrac 34 \cdot 2(\pi) (4)\\ &= 6\pi \end{align*}$$ This suggests that the radius of the cone's base ($r'$) is equal to: $$\begin{align*} C &= 2\pi r' \\ 6\pi &= 2\pi r' \\ r' &= 3. \end{align*}$$ Also, when we tape together the marked radii to form the cone, the radii in question become the slanted height of the cone. This can be used to find the actual height of the cone, which by the Pythagorean Theorem, is equal to: $$\begin{align*}SL^2 &= h^2 + (r')^2\\ 4^2 &= h^2 +3^2 \\ 16 &= h^2 + 9\\ 7 &= h^2 \\ h &= \sqrt{7}. \end{align*}$$ Thus, the volume of the cone is equal to: $$\begin{align*}V &= \dfrac 13 \pi r^2 h \\ &= \dfrac 13 \pi (r')^2 h\\ &= \dfrac 13 \pi (3)^2 \cdot \sqrt{7} \\ &= \dfrac 13 \pi 9\sqrt{7} \\ &= 3\pi\sqrt{7} \end{align*}$$

Thus, the correct answer is C .