Question

calculating probabilities an electronic dartboard has a diameter of 10 inches. the bull’s-eye has a radius of 1.5 inches. if a dart hits the board randomly, what is the chance it hits the bull’s-eye? check all that apply. the dartboard area is roughly 78.54 square inches. the dartboard area is roughly 314.16 square inches. the bull’s-eye area is roughly 7.07 square inches. the probability that a random dart strikes the bull’s-eye is roughly 0.09. the probability that a random dart strikes the bull’s-eye is roughly 0.02.

Explanation:

Step1: Calculate dartboard area

The dartboard has a diameter of 10 inches, so radius \( r_{dartboard} = \frac{10}{2} = 5 \) inches. The area of a circle is \( A = \pi r^2 \), so dartboard area \( A_{dartboard} = \pi \times 5^2 = 25\pi \approx 78.54 \) square inches.

Step2: Calculate bull's - eye area

The bull's - eye has a radius of 1.5 inches. Using the area formula \( A=\pi r^{2}\), bull's - eye area \( A_{bull - eye}=\pi\times(1.5)^{2}=2.25\pi\approx7.07 \) square inches.

Step3: Calculate probability of hitting bull's - eye

Probability \( P=\frac{A_{bull - eye}}{A_{dartboard}}=\frac{2.25\pi}{25\pi}=\frac{2.25}{25} = 0.09 \).

Now we check the options:

• "The dartboard area is roughly 78.54 square inches." - Correct (from Step1).
• "The dartboard area is roughly 314.16 square inches." - Incorrect (we calculated ~78.54).
• "The bull’s - eye area is roughly 7.07 square inches." - Correct (from Step2).
• "The probability that a random dart strikes the bull’s - eye is roughly 0.09." - Correct (from Step3).
• "The probability that a random dart strikes the bull’s - eye is roughly 0.02." - Incorrect (we calculated 0.09).

Answer:

• The dartboard area is roughly 78.54 square inches.
• The bull’s - eye area is roughly 7.07 square inches.
• The probability that a random dart strikes the bull’s - eye is roughly 0.09.