Question

Two circles of radius $1$ are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0, 0)$ to $(2, 0).$ The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0, 1)$ to $(2, 1).$ What is the probability that circles $A$ and $B$ intersect?

$\dfrac{2 + \sqrt{2}}{4}$

$\dfrac{3\sqrt{3} + 2}{8}$

$\dfrac{2\sqrt{2} - 1}{2}$

$\dfrac{2 + \sqrt{3}}{4}$

$\dfrac{4\sqrt{3} - 3}{4}$

Solution:

Let the centers be $(a, 0)$ and $(b, 1)$ with $a, b \in [0, 2].$ The circles (radius $1$ each) intersect iff the distance between centers is at most $2:$ $\sqrt{(a - b)^2 + 1} \le 2 \iff |a - b| \le \sqrt{3}.$

The pairs $(a, b)$ fill the square $[0, 2]^2$ of area $4.$ The failing region $|a - b| \gt \sqrt3$ is two right triangles, each with legs $2 - \sqrt3,$ of total area $(2 - \sqrt3)^2 = 7 - 4\sqrt3.$

So the favorable area is $4 - (7 - 4\sqrt3) = 4\sqrt3 - 3,$ and the probability is $\frac{4\sqrt3 - 3}{4}.$

Thus, the correct answer is E.

2008 AMC 12B Problem 21

Solution:

Problem 21 in Other Years