Question

question 3
a point in the figure below is chosen at
random. find the probability that the point
lies in the shaded region. give each answer
as a percent.
18 cm
answer:________________________

Explanation:

Step1: Calculate the radius of the circle

The diameter of each circle is \(18\div2 = 9\) cm, so the radius \(r = 9\div2=4.5\) cm.

Step2: Calculate the area of the rectangle

The length of the rectangle is \(l = 18\) cm, and the width \(w=9\) cm. The area of the rectangle \(A_{rectangle}=l\times w=18\times9 = 162\) \(cm^{2}\).

Step3: Calculate the area of the two circles

The area of a circle is \(A=\pi r^{2}\). For two circles, \(A_{circles}=2\times\pi r^{2}\). Substituting \(r = 4.5\) cm, we get \(A_{circles}=2\times\pi\times(4.5)^{2}=2\times\pi\times20.25 = 40.5\pi\approx40.5\times3.14 = 127.17\) \(cm^{2}\).

Step4: Calculate the area of the shaded region

\(A_{shaded}=A_{rectangle}-A_{circles}=162 - 127.17=34.83\) \(cm^{2}\).

Step5: Calculate the probability (as a percentage)

The probability \(P=\frac{A_{shaded}}{A_{rectangle}}\times100\%=\frac{34.83}{162}\times100\% = 21.5\%\).

Answer:

\(21.5\%\)