Problem:
If an arc of 45β on circle A has the same length as an arc of 30β on circle B, then the ratio of the area of circle A to the area of circle B is
Answer Choices:
A. 94β
B. 32β
C. 65β
D. 23β
E. 49β
Solution:
Let CAβ=2ΟRAβ be the circumference of circle A, let CBβ=2ΟRBβ be the circumference of circle B, and let L the common length of the two arcs. Then
36045βCAβ=L=36030βCBβ
Therefore
CBβCAββ=32β so 32β=2ΟRBβ2ΟRAββ=RBβRAββ
Thus, the ratio of the areas is
Area of Circle (B) Area of Circle (A)β=ΟRB2βΟRA2ββ=(RBβRAββ)2=94ββ
The problems on this page are the property of the MAA's American Mathematics Competitions