A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 54 feet and a height at the center of 12 feet.Find an equation of the semielliptical arch.
A) $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 729 } = 1$
B) $\frac { x ^ { 2 } } { 729 } - \frac { y ^ { 2 } } { 144 } = 1$
C) $\frac { x ^ { 2 } } { 729 } + \frac { y ^ { 2 } } { 144 } = 1$
D) $\frac { x ^ { 2 } } { 54 } + \frac { y ^ { 2 } } { 729 } = 1$
E) $\frac { x ^ { 2 } } { 144 } + \frac { y ^ { 2 } } { 729 } = 1$

Question

A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 54 feet and a height at the center of 12 feet.Find an equation of the semielliptical arch. 
A) $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 729 } = 1$
B) $\frac { x ^ { 2 } } { 729 } - \frac { y ^ { 2 } } { 144 } = 1$
C) $\frac { x ^ { 2 } } { 729 } + \frac { y ^ { 2 } } { 144 } = 1$
D) $\frac { x ^ { 2 } } { 54 } + \frac { y ^ { 2 } } { 729 } = 1$
E) $\frac { x ^ { 2 } } { 144 } + \frac { y ^ { 2 } } { 729 } = 1$

Quizplus · Accepted Answer

The equation of an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. Here, the major axis is 54 feet (half of which, $a$, is 27, so $a^2 = 729$), and the height at the center, which corresponds to the minor axis, is 12 feet ($b = 12$, so $b^2 = 144$). Therefore, the correct equation is $\frac{x^2}{729} + \frac{y^2}{144} = 1$.

A Semielliptical Arch Over a Tunnel for a One-Way Road x2144−y2729=1\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 729 } = 1144x2​−729y2​=1

A Semielliptical Arch Over a Tunnel for a One-Way Road $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 729 } = 1$