Question

11 multiple choice 2 points the diagram above shows a square with a side of 6. two concentric circles with radii of 1 and 2 have been placed inside the square. assuming a dart randomly strikes the interior region of the square, what is the probability that the dart lands inside the donut area between the two circles? 1/12 π 1/3 π 5/9 π 1/9 π

Explanation:

Step1: Calculate the area of the square

The area formula of a square is \(A = s^2\), where \(s\) is the side - length. Given \(s = 6\), then \(A_{square}=6\times6 = 36\).

Step2: Calculate the area of the two - circle region (donut area)

The area formula of a circle is \(A=\pi r^{2}\). The area of the larger circle with \(r = 2\) is \(A_{1}=\pi\times2^{2}=4\pi\), and the area of the smaller circle with \(r = 1\) is \(A_{2}=\pi\times1^{2}=\pi\). The area of the donut (annulus) is \(A_{donut}=A_{1}-A_{2}=4\pi-\pi = 3\pi\).

Step3: Calculate the probability

The probability \(P\) that the dart lands in the donut area is the ratio of the area of the donut to the area of the square. So \(P=\frac{A_{donut}}{A_{square}}=\frac{3\pi}{36}=\frac{1}{12}\pi\).

Answer:

\(\frac{1}{12}\pi\)