Solve the problem.
-Arches in the shapes of parabolas are often used in construction. Find the equation of a parabolic arc that is 18 feet high at its highest point and 26 feet wide at the base, as illustrated in the figure. Place the origin at the midpoint of the base.
A) $ y=-x^{2}+26 $
B) $ y=\frac{18}{169} x^{2}+18 $
C) $ y=-x^{2}+18 $
D) $ y=-\frac{18}{169} x^{2}+18 $

Question

Solve the problem.
-Arches in the shapes of parabolas are often used in construction. Find the equation of a parabolic arc that is 18 feet high at its highest point and 26 feet wide at the base, as illustrated in the figure. Place the origin at the midpoint of the base. 
A) $ y=-x^{2}+26 $ 
B) $ y=\frac{18}{169} x^{2}+18 $ 
C) $ y=-x^{2}+18 $ 
D) $ y=-\frac{18}{169} x^{2}+18 $

Quizplus · Accepted Answer

Since the origin is at the midpoint of the base, the parabolic arc is symmetric, which means we only need to focus on one side of the parabolic arc. We know that the vertex of a parabola is at the highest point, so the equation of the parabolic arc should be of the form $y = a(x-13)^2 + 18$, where $a$ is some constant.

We can now use the fact that the arc is 18 feet high at its highest point and 26 feet wide at the base to solve for $a$. At the highest point, $x=13$ and $y=18$, so we have $18 = a(13-13)^2 + 18 \Rightarrow a= \frac{18}{169}.$

Thus, the equation of the parabolic arc is $y = -\frac{18}{169} x^2 + 18$, which matches choice D.

Solve the Problem y=−x2+26 y=-x^{2}+26 y=−x2+26 B) y=18169x2+18 y=\frac{18}{169} x^{2}+18 y=16918​x2+18 C) y=−x2+18 y=-x^{2}+18 y=−x2+18

Solve the Problem $y=-x^{2}+26$ B) $y=\frac{18}{169} x^{2}+18$ C) $y=-x^{2}+18$