Intersections 26781

A rectangular grid consists of two mutually perpendicular systems of parallel lines with a distance of 2. We throw a circle with a diameter of 1 on this plane. Calculate the probability that this circle:

a) overlaps one of the straight lines;
b) do any of the intersections overlap the straight line?

Correct answer:

p₁ =  0.75
p₂ =  0.1963

Step-by-step explanation:

$$\begin{gathered} a=2 \\ D=1 \\ {S}_1=a^2=2^2=4 \\ {S}_2=\pi \cdot (D/2{)}^2=3.1416\cdot (1/2{)}^2\doteq 0.7854 \\ {S}_3=(a-D{)}^2=(2-1{)}^2=1 \\ {p}_1=\frac{S_1-S_3}{S_1}=\frac{4-1}{4}=\frac{3}{4}=0.75 \end{gathered}$$

$p_2=\frac{4\cdot \frac{S_2}{4}}{S_1}=\frac{4\cdot \frac{0.7854}{4}}{4}=0.1963$

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