To find the ratio of the areas of circle R and circle W, we first need to determine the ratio of their respective arc lengths.
The length of an arc on a circle is proportional to the measure of the central angle it subtends. In this case, we are given that the central angle on circle R is 90 degrees and the central angle on circle W is 60 degrees. Since these arcs have the same length, we can set up the following proportion:
Arc length on R / Arc length on W = Central angle on R / Central angle on W
Let's solve for the arc length ratio:
Arc length on R / Arc length on W = 90 degrees / 60 degrees
Simplifying this, we have:
Arc length on R / Arc length on W = 3/2
Now, the ratio of the areas of two circles is equal to the square of the ratio of their radii. Since the radius is half the diameter, we can also say the ratio of their areas is equal to the square of the ratio of their diameters.
Since the arc lengths are in the same ratio (3/2), we can conclude that the ratio of the diameters of circle R to circle W is also 3/2. Thus, the ratio of their areas will be (3/2)^2 = 9/4.
Therefore, the ratio of the area of circle R to the area of circle W is 9/4.