Question

determine the probability that a dart that lands on a random part of the target will land in the shaded scoring section. assume that all squares in the figure and all circles in the figure are congruent unless otherwise marked. round your answer to the nearest tenth of a percent, if necessary. sample problem the area of the square is 100 square inches. the area of the

Explanation:

Step1: Find the area of the circle

The radius of the circle $r = \frac{15}{2}$ inches. The area formula of a circle is $A_{circle}=\pi r^{2}$. So $A_{circle}=\pi(\frac{15}{2})^{2}=\frac{225\pi}{4}$ square - inches.

Step2: Find the area of the square

The diameter of the circle is equal to the length of the diagonal of the square. If the side - length of the square is $s$, using the Pythagorean theorem for the square ($d^{2}=s^{2}+s^{2}$, where $d = 15$ inches), we have $15^{2}=2s^{2}$, then $s^{2}=\frac{225}{2}$ square inches, and the area of the square $A_{square}=\frac{225}{2}$ square inches.

Step3: Calculate the probability

The probability $P$ that the dart lands in the shaded area (square) is given by the ratio of the area of the square to the area of the circle. $P=\frac{A_{square}}{A_{circle}}=\frac{\frac{225}{2}}{\frac{225\pi}{4}}=\frac{2}{\pi}\approx0.637$.

Step4: Convert to percentage

To convert the probability to a percentage, we multiply by 100. $P = 0.637\times100 = 63.7\%$.

Answer:

$63.7$