A new theme park is planning an archway at its main entrance. The arch is to be in the form of a semi-ellipse with the major axis as the span. If the span is to be 48 feet and the height at the center is to be 8 feet, what is the equation of the ellipse?

Question

Quizplus · Accepted Answer

The general equation for a semi-ellipse with major axis as the span and height at the center as the semi-minor axis is:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
where a is half of the span (in this case, 24) and b is half of the height at the center (in this case, 4).

Substituting the values, we get:
$$\frac{x^2}{24^2}+\frac{y^2}{4^2}=1$$
Simplifying further, we get:
$$\frac{x^2}{576}+\frac{y^2}{16}=1$$
Multiplying both sides by 576 to eliminate the fractions, we get:
$$16x^2+576y^2=576^2$$
which is in the form of the equation given in choice C.

A New Theme Park Is Planning an Archway at Its