Solve the problem.
-A bridge is built in the shape of a semi-elliptical arch. It has a span of 96 feet. The height of the arch 29 feet from the center is to be 6 feet. Find the height of the arch at its center rounded to the nearest hundredth of a foot.
A) 9.93 ft
B) 6.29 ft
C) 7.53 ft
D) 29.23 ft

Question

Quizplus · Accepted Answer

We can use the equation of a semi-ellipse to solve for the height of the arch at its center. The equation is:

y = b * sqrt(1 - x^2 / a^2)

where a is half of the span (96/2 = 48) and b is the height of the arch at 29 feet from the center (6). We can plug in these values and solve for y when x = 0 (the center of the arch):

y = 6 * sqrt(1 - 0^2 / 48^2) = 7.53

Therefore, the height of the arch at its center is approximately 7.53 feet, which is closest to choice C. We can use the equation of a semi-elliptical arch:
$y = a\sqrt{1-\frac{x^2}{b^2}}$

where $a$ is the height of the arch at its center, $b$ is half of the span, and $x$ is the horizontal distance from the center of the bridge. We can solve for $a$ by substituting the given values:

$6 = a\sqrt{1-\frac{(29)^2}{(96/2)^2}}$

$6 = a\sqrt{1-\frac{841}{2304}}$

$6 = a\sqrt{\frac{1463}{2304}}$

$a = \frac{6}{\sqrt{\frac{1463}{2304}}} \approx 7.53$

Therefore, the height of the arch at its center is approximately 7.53 feet, which corresponds to choice C.

Solve the Problem. -A Bridge Is Built in the Shape of a Semi-Elliptical