+77 In the diagram, two circles, each with center $D$, have radii of $1$ and $2$. The total area of the shaded region is $\frac5{12}$ of the area of the larger circle. How many degrees are in the measure of (the smaller) $\angle ADC$?
+129850 The area of the larger circle will be 4 pi
So...the shaded area is = (5/12) * 4 pi = 20/12 pi = ( 5/3 ) pi
Let the measure of the smaller angle ADC = theta (in rads)
And let the measure of the larger angle ADC = [2pi - theta ] in rads
So...the area of the sector subtended by the large circle is (1/2) 4^2 * θ = 2θ
And let the area of the sector subtended by the larger circle be (1/2) * 1^2 * (2pi - θ) =
(1/2) (2pi - θ)
So we have that
2θ + (1/2)(2pi - θ) = (5/3)pi
2θ + pi - θ/2 = (5/3)pi
(3/2)θ = 2/3 pi
θ = (2/3)(2/3) pi = (4/9) pi
So...the smaller angle ADC = (4/9)pi * 180 / pi = 20 * 4 = 80°
Proof
Area of shaded sector of lager circle = 2(4/9) pi = 8/9 pi
Area of shaded sector of smaller circle = (1/2) (2pi - (4/9) pi) = (1/2) (14/9) pi = (7/9) pi
Combined areas = (8/9 + 7/9) pi = (15/9) pi = (5/3) pi
