Question 684·Medium·Math

The radius is $12$ feet, so use the circle area formula, $A=\pi r^2$. Type $12^2$ in Desmos; it shows $144$. So $A=\pi(12)^2=144\pi\text{ square feet}$. Square the radius, not the diameter.

The cleaned sector has area $72\pi$ square feet out of the whole $144\pi$. Type $\frac{72\pi}{144\pi}$ on the next Desmos line; it shows $0.5$, so the sector covers one half of the circle. The sector's arc takes up that same fraction of the circle's circumference.

The circumference is the distance around the full circle. Use $C=2\pi r$ and type $2\cdot12$ in Desmos. It shows $24$, so $C=2\pi(12)=24\pi\text{ feet}$. This is the whole curved boundary, not yet the cleaned sector's arc.

Take half the circumference. Type $\frac12\cdot24$ in Desmos; it shows $12$. That's the number multiplying $\pi$ in the arc length: $\frac12(24\pi)=12\pi\text{ feet}$. So the cleaned sector's arc measures $12\pi$ feet. Choice C.