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: Calculate the area of the square target

The side - length of the square target is $s = 16$ inches. The area of a square is $A_{square}=s^{2}$. So, $A_{square}=16^{2}=256$ square inches.

Step2: Calculate the area of one circle

The diameter of each circle is $d = 8$ inches (since $16\div2 = 8$), and the radius $r=\frac{d}{2}=4$ inches. The area of a circle is $A_{circle}=\pi r^{2}$. So, $A_{circle}=\pi\times4^{2}=16\pi$ square inches.

Step3: Calculate the area of the four - circle shaded region

There are 4 congruent circles. The total area of the 4 - circle shaded region is $A_{shaded}=4\times A_{circle}=4\times16\pi = 64\pi$ square inches.

Step4: Calculate the probability

The probability $P$ that a dart lands in the shaded region is the ratio of the area of the shaded region to the area of the entire target. $P=\frac{A_{shaded}}{A_{square}}=\frac{64\pi}{256}=\frac{\pi}{4}\approx 0.7854$.

Step5: Convert the probability to a percentage

To convert the probability to a percentage, multiply by 100. $P(\text{percentage})=0.7854\times100 = 78.54\%\approx78.5\%$.

Answer:

$78.5$