We have to find the area of the 270 degree arc of the circumference.
The radius is 10 mi.
We know that the arc for the whole circumference is 360 degrees. Then, the area of the arc has to be a fraction of the total area of the circumference and this fraction is equal to the ratio of the arc, 270, to the whole circumference arc, 360.
Then, the fraction is:
[tex]\frac{270}{360}=0.75[/tex]
Then, we can calculate the area of the arc as 0.75 of the area of the circumference:
[tex]A=0.75A_c=0.75\cdot(\pi r^2)=0.75\cdot\pi\cdot10^2=0.75\cdot100\cdot\pi=75\pi\text{ mi}^2[/tex]
Answer: 75pi mi^2 (option C)
