Problem 1: Find the standard equation of an ellipse given that its center is at the origin $(0, 0)$, one vertex is at $(5, 0)$, and one focus is at $(3, 0)$.

Problem 2: Convert the general equation of the ellipse $9x^2 + 4y^2 - 36x + 24y + 36 = 0$ into standard form.

Problem 3: Find the foci, vertices, and eccentricity of the ellipse defined by $\frac{(x - 1)^2}{25} + \frac{(y + 2)^2}{16} = 1$.

Case Study 1: A whispering gallery is designed with a semi-elliptical ceiling. The maximum width of the room is $20\text{ m}$ (the major axis) and the maximum height of the ceiling at the center is $8\text{ m}$ (the semi-minor axis). Where should two people stand so they can perfectly hear each other's whispers?

Case Study 2: Calculate the length of the latus rectum and the total enclosed area of the ellipse defined by $\frac{x^2}{36} + \frac{y^2}{16} = 1$.