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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 $1.5.$

 Dec 23, 2023
 #1
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Since all rotations are the  same, let A be any point on  the circle

AC = 1.5  = AB

DA = 1 = DB

 

Using the Law of Cosines

BA^2 = DA^2 + DB^2   - 2 (DA * DB)cos (ADB)

1.5^2 = 1^2 - 1^2 - 2 ( 1 * 1) cos (ADB)

[ 2.25 - 1 -1 ]/ [ -2 ]  = cos (ADB)

.25 / -2 = cos (ADB)

-(1/8) = cos (ADB)

arccos (-1/8) ≈ 97.18°  =  ADB = ADC

 

Probability ≈   [  2 measure ADB ]   / 360  =   [2 * 97.18 ] / 360   ≈  54%

 

 

cool cool cool

 Dec 24, 2023

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