A $240^\circ$ sector of a circle of radius $15$ is rolled up (aligning its two straight sides) to form the lateral surface of a cone. What are the cone's slant height and radius?

1. The arc length of the $240^\circ$ sector of radius $15$ is $\frac{240}{360}\cdot2\pi(15)=20\pi$.
2. This arc becomes the base circumference, so $2\pi r=20\pi$, giving base radius $r=10$.
3. The cone's slant height equals the sector's radius, $15$. So the slant height is $15$ and the radius is $10$.