The radius of the smaller circle in the figure below is 6 units and the area of the overlapping rectangle is 243 sq units. How many units is the radius of the larger circle?
+44 This answer is going to assume that the short side of the rectange is equal to the radius of the large circle.
Our goal is to find the short side of the rectangle. The short side must be between 6 and 12(radius and diameter of small circle). Let's start by finding some factor pairs for 243.
Pair 1: 1*243 (doesn't work)
Pair 2: 3*81(doesn't work)
Pair 3: 9*27(works)
As those are the only three pairs, pair 3 will work.
Now that we know the short side is 9, we can see that the radius of the large circle is 9
+129930 The distance between the centers of the circles = 6 + R where R is the radius of the larger circle
So we can form a right triangle with the hypotenuse = (6 + R) and one leg = R - 6
So.....the width of the rectangle is sqrt [ (6 + R)^2 - ( R - 6)^2 ] = sqrt (24R)
And the height of the rectangle = R
So
sqrt ( 24R) * R = 243 square both sides
24R * R ^2 = 243^2
24R^3 = 243^2
R^3 = 243^2 / 24
R = (243^2 / 24)^(1/3) = 27/2 = 13.5
+1641 The radius of the smaller circle in the figure below is 6 units and the area of the overlapping rectangle is 243 sq units. How many units is the radius of the larger circle?
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A = 243 r = 6 R = ?
OQ = R + 6 PQ = R - 6 OP = 243 / R
(R + 6)2 - (R - 6)2 = (243 / R)2
R = 13.5 units 
