Solve the problem.
-A bridge is built in the shape of a semi-elliptical arch. It has a span of 118 feet. The height of the arch 30 feet from the center is to be 12 feet. Find the height of the arch at its center rounded to the nearest hundredth of a
Foot.
A)12.41 ft
B)23.6 ft
C)13.94 ft
D)30.64 ft

Question

Quizplus · Accepted Answer

Since the bridge is in the shape of a semi-elliptical arch, we can use the equation for the ellipse to find the height at the center. The equation is:
((h/2)^2)/(a^2/4) + (b^2/4) = 1
where h is the height at the center, a is the span/2, and b is the height 30 feet from the center. Substituting in the given values, we get:
((h/2)^2)/((118/2)^2/4) + (12^2/4) = 1
Simplifying and solving for h, we get:
h/2 = 13.944
h = 27.888 feet
Rounding to the nearest hundredth, the height at the center of the bridge is 13.94 feet. Therefore, the best choice is C. One way to approach the problem is to use the equation of an ellipse in standard form:
[(x-h)^2/a^2]+[(y-k)^2/b^2] = 1
where (h,k) is the center of the ellipse, a is the distance from the center to the horizontal axis (half of the span), and b is the distance from the center to the highest point of the arch (30 feet from the center). We know that a = 59 feet (half of the span), and b = 18 feet (30 feet from the center). We can plug these values into the equation and solve for y when x = 0 (at the center of the arch):
[(0-h)^2/59^2]+[(y-k)^2/18^2] = 1
Simplifying and solving for y, we get:
(y-k)^2/18^2 = 1-[(0-h)^2/59^2]
y-k = ±18*sqrt(1-[(0-h)^2/59^2])
y = k ± 18*sqrt(1-[(0-h)^2/59^2])
To find the height at the center of the arch, we need to plug in h = 0 (since the center is at the midpoint of the span) and round to the nearest hundredth:
y = 30 ± 18*sqrt(1-[(0-0)^2/59^2]) = 30 ± 18*sqrt(1-(0/3481)) = 30 ± 18 = 12 or 48
Since a height of 48 feet at the center of the arch would be unreasonable, the answer is 30-18 = 12 feet, rounded to the nearest hundredth. Therefore, the best choice is C) 13.94 ft.

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