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) $\angle ADC$?

codehtml127
Jun 28, 2018

#1**+1 **

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

CPhill
Jun 29, 2018