https://file.aiquickdraw.com/mathbot/file/1726016430085ajw8njs9.jpg

The problem from the image involves a semi-ellipse, where the arch of a bridge is described by its geometric properties.

**Given:**
- The arch is in the shape of a semi-ellipse.
- The major axis is along the water level and measures **120 ft** across (so the total length of the major axis is \( 2a = 120 \, \text{ft} \), meaning \( a = 60 \, \text{ft} \)).
- The arch is **20 ft high** in the middle (this is the distance from the center of the ellipse to the top of the arch, which is the **semi-minor axis** \( b = 20 \, \text{ft} \)).
- The goal is to find how high the arch is at a point **20 ft horizontally** from the center.

### Equation of the Ellipse:

For an ellipse centered at the origin, the general equation is:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Where:
- \( a = 60 \) (semi-major axis),
- \( b = 20 \) (semi-minor axis),
- \( x \) is the horizontal distance from the center,
- \( y \) is the vertical height above the water level at that point.

We are asked to find \( y \) when \( x = 20 \).

### Step 1: Plug values into the ellipse equation:
\[
\frac{20^2}{60^2} + \frac{y^2}{20^2} = 1
\]
Solve for \( y \).The height of the arch at a point 20 feet horizontally from the center is approximately **18.86 feet** above the water level.

Do you have any questions or would you like more details on this solution?

Here are 5 related questions:
1. How would the height change if the bridge arch was wider (e.g., 140 feet across)?
2. What is the equation of the ellipse if the arch is located 10 feet above the water level instead of at the water level?
3. Can you find the height of the arch at a point 40 feet horizontally from the center?
4. What would the semi-major and semi-minor axes be if the arch's height was 25 feet in the middle and 100 feet across?
5. How does changing the height in the middle of the ellipse affect the overall shape?

**Tip:** When solving problems involving ellipses, always ensure you are correctly identifying the semi-major and semi-minor axes, as they define the shape and dimensions of the ellipse.

The arch of the bridge is in the shape of a semi-ellipse, with its major axis at the water level. Suppose the arch is 20 ft high in the middle and 120 ft across the major axis. How high above the water level is the major arch at a point 20 ft (horizontally) from the center? Round off to 2 decimal places.

Learn how to calculate the height of an arch in the shape of a semi-ellipse. The arch is 120 ft wide and 20 ft high in the middle. How high is the arch at a point 20 ft horizontally from the center? Detailed step-by-step solution using the equation of an ellipse.

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