Two points on a circle of radius 1 are chosen at random. Find the probability that the distance between the two points is at most 1.5.
Let's denote the two points on the circle as A and B, and let's assume that A is fixed at the top of the circle (i.e., at the point (0,1)). We can then use polar coordinates to describe the position of point B on the circle. Let θ be the angle that the line segment AB makes with the positive x-axis, measured in radians.
We can then write the position of point B as (cos θ, sin θ), since it lies on the circle of radius 1 centered at the origin. Since we are choosing point B at random, the angle θ is uniformly distributed on the interval [0, 2π).
To find the probability that the distance between A and B is at most 1.5, we need to find the set of values of θ that satisfy this condition. The distance between A and B is given by the formula:
distance AB = sqrt((cos θ - 0)^2 + (sin θ - 1)^2) = sqrt(cos^2 θ + (sin θ - 1)^2)
So, the condition that the distance between A and B is at most 1.5 is equivalent to the inequality:
sqrt(cos^2 θ + (sin θ - 1)^2) ≤ 1.5
Squaring both sides and simplifying, we get:
cos^2 θ + (sin θ - 1)^2 ≤ 2.25
Expanding the second term and simplifying, we get:
sin^2 θ - 2sin θ + 2 ≤ 0
Using the quadratic formula to solve for sin θ, we get:
sin θ ≤ 1 ± sqrt(3) / 2
Since θ is uniformly distributed on the interval [0, 2π), the probability that sin θ satisfies this inequality is equal to the ratio of the length of the interval [0, 2π) for which sin θ satisfies this inequality to the length of the entire interval [0, 2π). The length of the interval [0, 2π) for which sin θ satisfies this inequality can be found by considering the two cases:
sin θ ≤ 1 + sqrt(3) / 2: In this case, we have:
0 ≤ θ ≤ pi/3 or 5pi/3 ≤ θ ≤ 2pi
The length of this interval is pi/3 + (2pi - 5pi/3) = 4pi/3.
sin θ ≤ 1 - sqrt(3) / 2: In this case, we have:
pi/3 ≤ θ ≤ 2pi/3 or 4pi/3 ≤ θ ≤ 5pi/3
The length of this interval is (2pi/3 - pi/3) + (5pi/3 - 4pi/3) = 2pi/3.
Therefore, the probability that the distance between A and B is at most 1.5 is:
(4pi/3 + 2pi/3) / 2pi = 1
So the probability that the distance between two randomly chosen points on the circle of radius 1 is at most 1.5 is 1 or 100%
