1. **Problem statement:** We need to find the time at 45 degrees west longitude when it is 2 am at 15 degrees east longitude. 2. **Formula and concept:** The Earth is divided into 360 degrees of longitude, and it rotates 360 degrees in 24 hours. Therefore, every 15 degrees corresponds to 1 hour difference in time. 3. **Calculate the longitude difference:** $$\text{Difference} = 15^\circ \text{E} - (-45^\circ \text{W}) = 15 + 45 = 60^\circ$$ 4. **Convert longitude difference to time difference:** $$\text{Time difference} = \frac{60^\circ}{15^\circ/\text{hour}} = 4 \text{ hours}$$ 5. **Determine the time at 45 degrees west:** Since 45 degrees west is west of 15 degrees east, the time there is earlier by 4 hours. 6. **Calculate the time:** $$2\text{ am} - 4 \text{ hours} = 10\text{ pm (previous day)}$$ **Final answer:** The time at 45 degrees west is 10 pm the previous day when it is 2 am at 15 degrees east.