1. **State the problem:** We have a rectangular board of dimensions 30 ft by 20 ft, with two circles inside it. The small circle has a diameter of 6 ft, and the large circle has a diameter of 10 ft. We want to find the probabilities of landing in each circle when an arrow is randomly thrown at the board. 2. **Formula for probability:** Probability of landing in a region = $ \frac{\text{Area of the region}}{\text{Area of the board}} $. 3. **Calculate the area of the board:** $$\text{Area}_{board} = 30 \times 20 = 600 \text{ ft}^2$$ 4. **Calculate the area of the small circle:** Diameter = 6 ft, so radius $r_s = \frac{6}{2} = 3$ ft. $$\text{Area}_{small} = \pi r_s^2 = \pi \times 3^2 = 9\pi \approx 28.27 \text{ ft}^2$$ 5. **Calculate the area of the large circle:** Diameter = 10 ft, so radius $r_l = \frac{10}{2} = 5$ ft. $$\text{Area}_{large} = \pi r_l^2 = \pi \times 5^2 = 25\pi \approx 78.54 \text{ ft}^2$$ 6. **Calculate the probability of landing in the small circle:** $$P_{small} = \frac{\text{Area}_{small}}{\text{Area}_{board}} = \frac{9\pi}{600} = \frac{\cancel{3} \times 3 \pi}{\cancel{3} \times 200} = \frac{3\pi}{200} \approx \frac{3 \times 3.1416}{200} = \frac{9.4248}{200} = 0.0471$$ 7. **Calculate the probability of landing in the large circle:** $$P_{large} = \frac{\text{Area}_{large}}{\text{Area}_{board}} = \frac{25\pi}{600} = \frac{\cancel{25} \pi}{\cancel{25} \times 24} = \frac{\pi}{24} \approx \frac{3.1416}{24} = 0.1309$$ 8. **Calculate the probability of landing in either circle:** $$P_{either} = P_{small} + P_{large} = 0.0471 + 0.1309 = 0.178 \approx 0.18$$ 9. **Compare with the options:** - A. Probability of landing in a circle is approximately 0.18. **True** - B. Probability of landing in the small circle is approximately 0.03. **False** (calculated 0.0471) - C. Probability of landing in the large circle is approximately 0.05. **False** (calculated 0.1309) - D. Probability of landing in the large circle is approximately 0.13. **True** - E. Probability of landing in the small circle is approximately 0.19. **False** **Final answers:** A and D are true.