Question

competency 16 (part 2): mathematical communication

1. in the diagram of a target shown to the right, each of the concentric rings numbered 10, 20, 30, or 40 has a radius that is 1\ larger than the next smallest ring. the bullseye, worth 50 points, has a 1\ radius.

marissa is a skillful dart player. what is the probability that her dart will land on the 30-ring?

a. 40%
b. 30%
c. 20%
d. 10%

Explanation:

⚡ Using what you learned: Density, Mass, and Geometric Probability

Step 1: Determine the radii of the concentric circles

The target consists of concentric circles starting from the center (bullseye) outward:

• Bullseye (\(50\) points): radius \(r_1 = 1\text{ inch}\)
• \(40\)-ring: radius \(r_2 = 1 + 1 = 2\text{ inches}\)
• \(30\)-ring: radius \(r_3 = 2 + 1 = 3\text{ inches}\)
• \(20\)-ring: radius \(r_4 = 3 + 1 = 4\text{ inches}\)
• \(10\)-ring: radius \(r_5 = 4 + 1 = 5\text{ inches}\)

Step 2: Calculate the total area of the target

The total area is the area of the outermost circle with radius \(r_5 = 5\text{ inches}\):

$$ A_{\text{total}} = \pi \cdot 5^2 = 25\pi $$

Step 3: Calculate the area of the 30-ring

The \(30\)-ring is the region between the circle of radius \(r_3 = 3\text{ inches}\) and the circle of radius \(r_2 = 2\text{ inches}\):

$$ A_{30\text{-ring}} = \pi \cdot 3^2 - \pi \cdot 2^2 = 9\pi - 4\pi = 5\pi $$

Step 4: Calculate the geometric probability

The probability that the dart lands on the \(30\)-ring is the ratio of the area of the \(30\)-ring to the total area:

$$ P = \frac{A_{30\text{-ring}}}{A_{\text{total}}} = \frac{5\pi}{25\pi} = \frac{1}{5} = 0.20 = 20\% $$

Answer:

C. 20%