Question

guided practice
the graphic shows a dartboard with four point ranges. the width of each ring is equal to the radius of the inner circle.
what is the probability of scoring at least 10 points?
a. 45%
b. 30%
c. 10%
d. 25%

Explanation:

Step1: Define inner circle radius

Let the radius of the inner circle (20-point area) be $r$.

Step2: Find total dartboard radius

Each ring has width $r$, so the outermost radius is $r + r + r + r = 4r$.

Step3: Calculate total dartboard area

Use circle area formula $A=\pi R^2$:
$A_{\text{total}} = \pi (4r)^2 = 16\pi r^2$

Step4: Calculate area for ≥10 points

The area for at least 10 points is the area of the two inner circles (10 and 20 points). The radius of the outer boundary of this region is $r + r = 2r$:
$A_{\text{≥10}} = \pi (2r)^2 = 4\pi r^2$

Step5: Compute probability

Probability is the ratio of the two areas:
$\text{Probability} = \frac{A_{\text{≥10}}}{A_{\text{total}}} = \frac{4\pi r^2}{16\pi r^2} = \frac{1}{4} = 25\%$

Answer:

D. 25%