Problem:
If an arc of 45∘ on circle A has the same length as an arc of 30∘ on circle B, then the ratio of the area of circle A to the area of circle B is
Answer Choices:
A. 94
B. 32
C. 65
D. 23
E. 49
Solution:
Let CA=2πRA be the circumference of circle A, let CB=2πRB be the circumference of circle B, and let L the common length of the two arcs. Then
36045CA=L=36030CB
Therefore
CBCA=32 so 32=2πRB2πRA=RBRA
Thus, the ratio of the areas is
Area of Circle (B) Area of Circle (A)=πRB2πRA2=(RBRA)2=94
The problems on this page are the property of the MAA's American Mathematics Competitions