Question

4 multiple choice 1 point
the diagram above shows a square with a side of 6. two concentric circles with radii of 1 and 3 have been placed inside the square. assuming a dart randomly strikes the interior region of the square, approximately what is the probability that the dart does not land inside any of the circles?
0.30
0.25
0.20
0.15

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 \(A_{square}=6^2 = 36\).

Step2: Calculate the area of the two circles

The area formula of a circle is \(A=\pi r^{2}\).
For the circle with \(r = 3\), \(A_{1}=\pi\times3^{2}=9\pi\).
For the circle with \(r = 1\), \(A_{2}=\pi\times1^{2}=\pi\).
The total area of the two circles is \(A_{circles}=A_{1}-A_{2}=9\pi-\pi = 8\pi\approx8\times3.14 = 25.12\).

Step3: Calculate the probability

The probability \(P\) that the dart does not land inside any of the circles is \(P=\frac{A_{square}-A_{circles}}{A_{square}}\).
Substitute \(A_{square}=36\) and \(A_{circles}\approx25.12\) into the formula: \(P=\frac{36 - 25.12}{36}=\frac{10.88}{36}\approx0.30\).

Answer:

0.30