Question

a dart is thrown randomly and lands within the square below. find the probability that the dart will land within the shaded area. the square has side lengths of 31, the smallest circle has a radius of 3, and the distance between each concentric circle is 3. round to the nearest hundredth.

a. 0.15
b. 0.26
c. 0.3
d. 0.44

Explanation:

Step1: Calculate area of the square

The area formula for a square is $A = s^2$, where $s$ is the side - length. Given $s = 31$, so $A_{square}=31^2=961$.

Step2: Calculate radius of the largest circle

The smallest circle has a radius $r_1 = 3$, and the distance between each concentric circle is 3. The radius of the largest circle $r_2=3 + 3+3=9$.

Step3: Calculate area of the largest circle

The area formula for a circle is $A=\pi r^{2}$. So $A_{circle}=\pi\times9^{2}=81\pi\approx81\times3.14 = 254.34$.

Step4: Calculate the probability

The probability $P$ that the dart lands in the shaded area (the circle) is the ratio of the area of the circle to the area of the square. $P=\frac{A_{circle}}{A_{square}}=\frac{254.34}{961}\approx0.26$.

Answer:

B. 0.26