1. **Problem 1: Find the foci of the ellipse given by** $$25x^2 + 16y^2 = 400$$. 2. **Rewrite the ellipse equation in standard form:** $$\frac{x^2}{\frac{400}{25}} + \frac{y^2}{\frac{400}{16}} = 1 \implies \frac{x^2}{16} + \frac{y^2}{25} = 1$$ 3. **Identify the ellipse parameters:** - $a^2 = 25$ (larger denominator), so $a = 5$ - $b^2 = 16$, so $b = 4$ 4. **Since $a^2 > b^2$, the major axis is vertical along the y-axis.** 5. **Calculate the focal distance $c$ using:** $$c = \sqrt{a^2 - b^2} = \sqrt{25 - 16} = \sqrt{9} = 3$$ 6. **The foci are located at:** $$(0, \pm c) = (0, 3) \text{ and } (0, -3)$$ 7. **Answer for Problem 1:** The foci are at $(0, 3)$ and $(0, -3)$. --- 8. **Problem 2: Find the maximum distance between Lucas and Neal walking on circular paths.** 9. **Lucas's path is a circle centered at $(-6, 5)$ with radius $6$:** $$(x + 6)^2 + (y - 5)^2 = 36$$ 10. **Neal's path is another circle (details not fully given), but to find maximum distance between walkers, assume Neal is on a circle centered at some point with radius $r$ (unknown).** 11. **Maximum distance between two points on two circles is the sum of the distance between centers plus both radii.** 12. **Since Neal's circle is not given, we cannot compute exact maximum distance without more info.** 13. **If Neal's circle center and radius were known, the formula would be:** $$\text{max distance} = \text{distance between centers} + r_1 + r_2$$ 14. **Without Neal's circle info, the maximum distance cannot be determined.** --- **Final answers:** - Foci of ellipse: $(0, 3)$ and $(0, -3)$ - Maximum distance between walkers: Insufficient information to determine.