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A circle of radius 1 is concentric with circles of radii $$a$$ and $$b$$ with $$b > a > 1$$. The area of the smallest circle equals the area of the ring with outer radius $$a$$ and inner radius 1 and also equals the area of the ring with outer radius $$b$$ and inner radius $$a$$. Then $$a + b$$ is ___________
Correct Answer: 3.146
The smallest circle has area $$\pi$$. So $$\pi(a^2 - 1) = \pi$$ gives $$a^2 = 2$$, and $$\pi(b^2 - a^2) = \pi$$ gives $$b^2 = 3$$. Hence $$a + b = \sqrt{2} + \sqrt{3} \approx 3.146$$.
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