If an arc of 60° on circle I has the same length as an arc of 45° on circle II, the ratio of the area of circle I to that of circle II is:

1968 AMC 12 Problem 11

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11.

If an arc of

60^\circ

on circle

\mathrm{I}

has the same length as an arc of

45^\circ

on circle

\mathrm{II},

the ratio of the area of circle

\mathrm{I}

to that of circle

\mathrm{II}

is:

16:9

9:16

4:3

3:4

none of these

Equate

(\frac{\theta}{360^\circ})2\pi r

for the two arcs

Square the resulting radius ratio to get the area ratio

Equal arc lengths give

60r_1=45r_2,

\frac{r_1}{r_2}=\frac{3}{4}.

Circle areas scale as the squares of the radii, giving

A_1:A_2=9:16.

Therefore, the correct answer is B.