Question

lance painted an 18 - inch - square board with an inscribed circle, as shown in the diagram. he plans to use it as a dart board. 18 inches a dart is dropped above the dart board. the dart has an equal chance of landing on any spot on the board. what is the probability that the dart lands in the shaded region of the dart board? round the answer to the nearest thousandths. enter the answer in the box.

Explanation:

Step1: Calculate area of square

The area formula for a square is $A = s^2$, where $s = 18$ inches. So $A_{square}=18^2 = 324$ square - inches.

Step2: Calculate area of circle

The diameter of the inscribed circle is equal to the side - length of the square, so $d = 18$ inches and $r=\frac{d}{2}=9$ inches. The area formula for a circle is $A=\pi r^2$. So $A_{circle}=\pi\times9^2 = 81\pi\approx81\times3.14159 = 254.469$ square - inches.

Step3: Calculate area of shaded region

$A_{shaded}=A_{square}-A_{circle}=324 - 254.469 = 69.531$ square - inches.

Step4: Calculate probability

The probability $P$ that the dart lands in the shaded region is the ratio of the area of the shaded region to the area of the square. So $P=\frac{A_{shaded}}{A_{square}}=\frac{69.531}{324}\approx0.215$.

Answer:

$0.215$