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.
since the arc lengths are the same,
90r = 60w
So, r/w = 2/3
Since the areas vary as the square of the radii,
R/W = = (r/w)^2 = 4/9
Steve is right,
I can't be bothered to find my error in all my mess.
360 / 60 = 6
Yes
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.
let the radius of circle R be r
let the radius of circle W be w
recall that arclength = rØ, where Ø is the angle in radians
arc length on first circle = (π/2)r
arc length of 2nd circle = (π/3)w
but they are equal
πr/2 = πw/3
r = 3w/2
area of first sector = (1/4)πr^2
= (1/4)π(9w^2)/4
= (9/16)πw^2
area of 2nd sector = (1/6)π w^2
ratio of first area : second area
= (9/16)π w^2 : (1/6)π w^2
= 9/16 : 1/6
= 54 : 16
= 27 : 8
or 27/8 as a fraction
check my arithmetic