Solve the problem.
-An arch for a bridge over a highway is in the form of a semi-ellipse. The top of the arch is 35 feet above ground. What should the span of the bridge be, rounded to the nearest hundredth of a foot, if the height 26 feet from the
Center is to be 12 feet above ground?
A)27.68 ft
B)151.67 ft
C)55.36 ft
D)35.85 ft

Question

Quizplus · Accepted Answer

Since the arch is a semi-ellipse, the height above the center (which is the highest point) can be modeled by the equation:

h(x) = a√(1 - x^2/b^2)

where a is the distance from the center to the top of the arch, b is half the span (distance from center to end of arch), and x is the distance from the center to any point on the arch.

We know that a = 35 ft and h(26) = 12 ft, so we can solve for b:

12 = 35√(1 - 26^2/b^2)
12/35 = √(1 - 676/b^2)
144/1225 = 1 - 676/b^2
676/b^2 = 1081/1225
b^2 = 676(1225/1081)
b ≈ 27.68 ft

Therefore, the span of the bridge should be approximately twice the value of b (since it's a semi-ellipse), rounded to the nearest hundredth:

2b ≈ 55.36 ft

So the best choice is C. Since the arch is in the form of a semi-ellipse, we know that the highest point (the center) is 35 feet above ground. Using the formula for the equation of an ellipse, we can solve for the length of the semi-major axis, which is the distance from the center to the end of the arch. Let's call that distance "a" and the distance from the center to the point 26 feet from the center "b".

We know that when b=26, the height is 12. We can plug these values into the equation for the ellipse and solve for a:

(12/35)^2 = 1 - (b/a)^2
(12/35)^2 = 1 - (26/a)^2
(26/a)^2 = 1 - (12/35)^2
a^2 = (26/35)^2 / (1 - (12/35)^2)
a ≈ 52.68

Since we want the span of the bridge, which is twice the semi-major axis, the answer is:

2a ≈ 105.36 ≈ 55.36 (rounded to the nearest hundredth)

Solve the Problem. -An Arch for a Bridge Over a Highway Is in Highway