a bridge is to have a span of 120 feet The height of the arch at a distance of 40 feet from the center is to be 6 feet Find the height of the arch at its center

Question

A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 6 feet. Find the height of the arch at its center

Answer 1

The standard form for the equation of an ellipse with enter (h, k), horizontal major axis of length 2a, and minor axis of length 2b is:

 

(x-h)²/a² + (y-k)²/b² = 1

 

For simplicity, let's assume that (h,k) = (0, 0), then the equation becomes x²/a² + y²/b² = 1.

 

Since the span is 120, 2a = 120, so a = 60.

 

We then have x²/3600 + y²/b² = 1

 

When x = 40, y = 6.  So, 1600/3600 + 36/b² = 1`

 

                                                      36/b² = 5/9

 

                                                       5b² = 324

 

                                                         b² = 324/5

 

Height of arch at the center = b = √(324/5) ≈ 8.05 ft