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 square is 100 square inches. enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Recall probability formula for area - based problems

The probability $P$ that a dart lands in a shaded region is given by $P=\frac{\text{Area of shaded region}}{\text{Total area}}$.

Step2: Calculate area of shaded region and total area for the first - figure

For the first figure, the side of the square is $10$ inches, so the area of the square $A_{square}=10\times10 = 100$ square inches. The diameter of the circle is equal to the side of the square, so $d = 10$ inches and the radius $r=5$ inches. The area of the circle $A_{circle}=\pi r^{2}=\pi\times5^{2}=25\pi\approx 25\times 3.14 = 78.5$ square inches. Then $P_1=\frac{A_{circle}}{A_{square}}=\frac{78.5}{100}=0.785\approx79\%$.

Step3: Calculate area of shaded region and total area for the second - figure

For the second figure, the hexagon can be divided into six equilateral triangles. The side length of the hexagon $a = 10$ inches. The area of an equilateral triangle with side length $a$ is $A_{\triangle}=\frac{\sqrt{3}}{4}a^{2}$. The area of the hexagon $A_{hexagon}=6\times\frac{\sqrt{3}}{4}a^{2}=6\times\frac{\sqrt{3}}{4}\times10^{2}=150\sqrt{3}\approx150\times1.732 = 259.8$ square inches. The radius of the circle $r = 10$ inches, and the area of the circle $A_{circle}=\pi r^{2}=\pi\times10^{2}=100\pi\approx314$ square inches. Then $P_2=\frac{A_{hexagon}}{A_{circle}}=\frac{259.8}{314}\approx0.83 = 83\%$.

Answer:

For the first figure: $79\%$; For the second figure: $83\%$