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The problem describes the arch of a bridge in the shape of a semi-ellipse, with a major axis along the water level. The key information provided is as follows:

- The height of the arch in the middle (i.e., the vertex) is 20 feet.
- The major axis is 120 feet across (which means the total width at the water level is 120 feet).
- We are asked to find the height of the arch at a point 20 feet horizontally from the center.

### Semi-Ellipse Equation

For an ellipse centered at the origin with a major axis along the \(x\)-axis, the standard equation is:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

Where:
- \(a\) is the semi-major axis (half of the total width),
- \(b\) is the semi-minor axis (the height of the arch at the center).

In this case:
- \(a = \frac{120}{2} = 60\) feet (half of the major axis),
- \(b = 20\) feet (the height at the center).

### Substituting into the Equation

The equation of the semi-ellipse becomes:

\[
\frac{x^2}{60^2} + \frac{y^2}{20^2} = 1
\]

At the point 20 feet horizontally from the center (\(x = 20\)), we substitute \(x = 20\) into the equation to find \(y\) (the height above the water level).

\[
\frac{20^2}{60^2} + \frac{y^2}{20^2} = 1
\]

Now, let's solve this equation for \(y\).The height of the arch 20 feet horizontally from the center is approximately 18.86 feet above the water level.

Would you like more details on any part of the solution or have any other questions?

Here are five related questions for further practice:
1. What would the height be if the horizontal distance from the center were 30 feet?
2. How would the equation change if the total width of the arch were 150 feet instead of 120 feet?
3. How would the height change if the semi-minor axis were increased to 25 feet while keeping the same width?
4. What is the height of the arch at the edge, 60 feet horizontally from the center?
5. How can you model this problem using a parabola instead of an ellipse?

**Tip:** Understanding how the semi-ellipse equation relates to different points on the arch helps with calculating heights and distances accurately.

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 at a given horizontal distance from the center using the equation of a semi-ellipse. This problem involves substituting values into the ellipse equation to find the height of a bridge arch. The arch is 20 feet high at the center and spans 120 feet across. Suitable for high school students.

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