1. **Problem statement:** Find the first day $t$ after the spring equinox where the day length $L(t)$ equals 750 minutes, given: $$L(t) = 52 \sin\left(\frac{2\pi}{365}t\right) + 728$$ where $t$ is in radians and days after the spring equinox. 2. **Set up equation:** We want to solve for $t$ when: $$L(t) = 750$$ Substitute $L(t)$: $$52 \sin\left(\frac{2\pi}{365}t\right) + 728 = 750$$ 3. **Isolate the sine term:** $$52 \sin\left(\frac{2\pi}{365}t\right) = 750 - 728 = 22$$ 4. **Divide both sides by 52:** $$\sin\left(\frac{2\pi}{365}t\right) = \frac{22}{52} \approx 0.4231$$ 5. **Find the principal solution for the angle:** Let $$x = \frac{2\pi}{365}t$$ Then: $$\sin(x) = 0.4231$$ The principal angle in radians is: $$x = \arcsin(0.4231) \approx 0.436$$ 6. **Find the first positive solution for $t$:** Since sine is positive in the first and second quadrants: $$x_1 = 0.436$$ $$x_2 = \pi - 0.436 = 2.705$$ Corresponding $t$ values are: $$t_1 = \frac{365}{2\pi} x_1 = \frac{365}{6.283} \times 0.436 \approx 25.34$$ $$t_2 = \frac{365}{2\pi} x_2 = \frac{365}{6.283} \times 2.705 \approx 157.17$$ 7. **First day after spring equinox:** The first time the day length reaches 750 minutes after the spring equinox is at $t_1 \approx 25.34$ days. 8. **Rounding:** Rounded to the nearest whole day: $$t \approx 25$$ **Final answer:** The first day after the spring equinox when the day length is 750 minutes is day **25**.