If a central angle of $90$ degrees defines an arc on circle $R$ that has the same length as the arc on circle $W$ defined by a $60$-degree central angle, what is the ratio of the area of circle $R$ to the area of circle $W$? Express your answer as a common fraction.
Arc Length = [90 x pi/180 x R] =[60 x pi/180 x W], solve for R/W
R/W = 2/3 - ratio of the area of circle R to circle W
+128474 Let R1 be the radius of the first circle
And let R2 be the measure of the second circle
So
Arc length = radius * theta ( in rads )
So....since the arc lenghts are equal....
R1 * pi/2 = R2 * pi/3
R1 / 2 = R2 / 3
R1 = (2/3)R2
So...the area of the first circle is
[ pi * (R1) ^2 ] = pi *[ (2/3) R2 ] ^2 = pi (4/9) (R2)^2
And the area of the second circle is
pi [ R2]^2
So the ratio of the area of the first cirlce to the second is
[ pi * (4/9) (R2)^2 ] 4
_______________ = ____
[ pi * (R2)^2 ] 9
