1. **State the problem:** We are given the function $$d(t) = 3 \sin\left(\frac{2\pi}{365}(t - 79)\right) + 12$$ which models the number of hours of daylight on day $$t$$ of the year. We need to find which day among 9, 60, 170, and 353 has the number of daylight hours closest to 11. 2. **Recall the formula:** $$d(t) = 3 \sin\left(\frac{2\pi}{365}(t - 79)\right) + 12$$ 3. **Evaluate $$d(t)$$ for each given day:** - For $$t=9$$: $$d(9) = 3 \sin\left(\frac{2\pi}{365}(9 - 79)\right) + 12 = 3 \sin\left(\frac{2\pi}{365}(-70)\right) + 12$$ Calculate the angle: $$\theta = \frac{2\pi}{365} \times (-70) = -\frac{140\pi}{365}$$ - For $$t=60$$: $$d(60) = 3 \sin\left(\frac{2\pi}{365}(60 - 79)\right) + 12 = 3 \sin\left(\frac{2\pi}{365}(-19)\right) + 12$$ Calculate the angle: $$\theta = \frac{2\pi}{365} \times (-19) = -\frac{38\pi}{365}$$ - For $$t=170$$: $$d(170) = 3 \sin\left(\frac{2\pi}{365}(170 - 79)\right) + 12 = 3 \sin\left(\frac{2\pi}{365}(91)\right) + 12$$ Calculate the angle: $$\theta = \frac{2\pi}{365} \times 91 = \frac{182\pi}{365}$$ - For $$t=353$$: $$d(353) = 3 \sin\left(\frac{2\pi}{365}(353 - 79)\right) + 12 = 3 \sin\left(\frac{2\pi}{365}(274)\right) + 12$$ Calculate the angle: $$\theta = \frac{2\pi}{365} \times 274 = \frac{548\pi}{365}$$ 4. **Calculate sine values and daylight hours:** Using approximate values for sine: - $$\sin\left(-\frac{140\pi}{365}\right) \approx \sin(-1.204) \approx -0.936$$ - $$\sin\left(-\frac{38\pi}{365}\right) \approx \sin(-0.327) \approx -0.321$$ - $$\sin\left(\frac{182\pi}{365}\right) \approx \sin(1.567) \approx 0.999$$ - $$\sin\left(\frac{548\pi}{365}\right) = \sin\left(\frac{548\pi}{365} - 2\pi\right) = \sin(1.567 - 6.283) = \sin(-4.716) \approx 0.999$$ Calculate daylight hours: - $$d(9) = 3 \times (-0.936) + 12 = -2.808 + 12 = 9.192$$ - $$d(60) = 3 \times (-0.321) + 12 = -0.963 + 12 = 11.037$$ - $$d(170) = 3 \times 0.999 + 12 = 2.997 + 12 = 14.997$$ - $$d(353) = 3 \times 0.999 + 12 = 2.997 + 12 = 14.997$$ 5. **Compare to 11 hours:** - Day 9: 9.192 hours (difference 1.808) - Day 60: 11.037 hours (difference 0.037) - Day 170: 14.997 hours (difference 3.997) - Day 353: 14.997 hours (difference 3.997) 6. **Conclusion:** Day 60 has the number of daylight hours closest to 11. **Final answer:** Day 60