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. the area of the square is 100 square inches. the area of the

Explanation:

Step1: Calculate area of the circle in the first - part

The diameter of the circle is equal to the side - length of the square, so $d = 10$ in, and the radius $r=\frac{d}{2}=5$ in. The area formula for a circle is $A=\pi r^{2}$. Substituting $r = 5$ in, we get $A_1=\pi\times5^{2}=25\pi\approx 25\times3.14 = 78.5$ square inches. The area of the square is $A_{s1}=100$ square inches. The probability $P_1$ that a dart lands in the shaded (circle) region is $P_1=\frac{A_1}{A_{s1}}=\frac{25\pi}{100}=\frac{\pi}{4}\approx 0.785$. As a percentage, $P_1 = 78.5\%$.

Step2: Calculate area of the shaded region in the second - part

The area of the outer - circle with radius $R = \frac{15}{2}=7.5$ in is $A_{o}=\pi R^{2}=\pi\times(7.5)^{2}=56.25\pi$ square inches. The area of the inner - circle with radius $r = 5$ in is $A_{i}=\pi r^{2}=\pi\times5^{2}=25\pi$ square inches. The area of the shaded region $A_2=A_{o}-A_{i}=\pi(56.25 - 25)=31.25\pi\approx31.25\times3.14 = 98.125$ square inches. The area of the square is $A_{s2}=15\times15 = 225$ square inches. The probability $P_2$ that a dart lands in the shaded region is $P_2=\frac{A_2}{A_{s2}}=\frac{31.25\pi}{225}=\frac{31.25\times3.14}{225}\approx\frac{98.125}{225}\approx0.436$. As a percentage, $P_2\approx43.6\%$.

Answer:

43.6%