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### Problem 5: Semielliptical Arch Bridge
- **Given**: The bridge is modeled by the upper half of an ellipse. The width of the river (and hence the span of the ellipse along the x-axis) is 20 meters, and the height of the arch at the center is 6 meters above the river (y-axis).
- **Task**: Find the equation of the ellipse.

1. The general equation of an ellipse is:
   \[
   \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
   \]
   where \(2a\) is the width (20 meters), so \(a = 10\), and \(b\) is the height (6 meters).

2. Substituting the values:
   \[
   \frac{x^2}{10^2} + \frac{y^2}{6^2} = 1
   \]
   Simplified:
   \[
   \frac{x^2}{100} + \frac{y^2}{36} = 1
   \]
   
   This is the equation of the ellipse that describes the bridge's arch.

---

### Problem 6: Whispering Gallery
- **Given**: The length of the hall is 100 feet, so the total distance along the major axis is 100 feet. The foci are located 25 feet from the center.
- **Task**: Find the height of the ceiling at the center of the hall.

1. The equation of an ellipse is the same:
   \[
   \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
   \]
   Here, the length of the hall gives \(2a = 100\), so \(a = 50\).

2. The distance from the center to each focus is given as 25 feet. This distance is represented by \(c\), and in an ellipse, the relationship between \(a\), \(b\), and \(c\) is:
   \[
   c^2 = a^2 - b^2
   \]
   Substituting \(a = 50\) and \(c = 25\):
   \[
   25^2 = 50^2 - b^2
   \]
   \[
   625 = 2500 - b^2
   \]
   Solving for \(b^2\):
   \[
   b^2 = 2500 - 625 = 1875
   \]
   Therefore:
   \[
   b = \sqrt{1875} \approx 43.30 \text{ feet}
   \]

   So, the height of the ceiling at the center is approximately **43.30 feet**.

---

Do you need further clarifications, or do you want to discuss these solutions in more detail?

Here are 5 related questions to expand your understanding:
1. How would the equation of the bridge's arch change if the span of the bridge increased to 30 meters?
2. What would happen to the height of the whispering gallery if the foci were located 35 feet from the center instead of 25 feet?
3. Could the same equation form be used if the arch were a parabola instead of an ellipse?
4. How does changing the height of the arch affect the structural strength of a bridge?
5. What are other real-world applications of elliptical shapes in architecture?

**Tip**: Always identify the major and minor axes clearly when working with ellipses; this simplifies setting up the equation significantly.

1. Semielliptical Arch Bridge: An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river 20 meters wide. The center of the arch is 6 meters above the center of the river. Write an equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch.

2. Whispering Gallery: A hall 100 feet in length is to be designed as a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center? Round your answer to the nearest hundredth.

Solve problems involving a semielliptical arch bridge spanning 20 meters with a height of 6 meters and a whispering gallery 100 feet in length with foci located 25 feet from the center. Learn to find the equation of an ellipse and determine the ceiling height at the center using the properties of ellipses.

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