Question 608·Medium·Math

Arc $BC$ borders the sector with $\angle BOC$, the second angle in $1:2:3$, so it gets $2$ ratio parts. The three sectors fill the circle, so the whole is all the ratio parts added together. Its fraction $f$ of the circle is $f=\frac{2}{1+2+3}$. Using $\frac{2}{3}$ would compare this angle with the third angle, not with the whole circle.

Type $f=\frac{2}{1+2+3}$ in Desmos, then type $f$ on its own line. Desmos shows $f=0.333\ldots$ under PARAMETERS, and the $f$ line shows $0.333\ldots$; click the fraction button on that line to see $\frac13$. So the angle for arc $BC$ takes one-third of the full turn. Its arc takes the same fraction of the circumference.

The circumference is $120\pi$. Add $120f$ in Desmos to multiply its coefficient by the arc’s fraction. Desmos shows $40$. Keep the $\pi$ from the circumference, so the length of arc $BC$ is $40\pi$. Choice B.