Solve the problem.
-A bridge is built in the shape of a semielliptical arch. It has a span of 92 feet. The height of the arch 30 feetfrom the center is to be 11 feet. Find the height of the arch at its center.
A)14.51 ft
B)11.64 ft
C)30.9 ft
D)16.87 ft

Question

Quizplus · Accepted Answer

The equation of an ellipse is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. In this case, the span of the bridge (92 feet) represents the length of the major axis, so $a = 46$ feet. The height of the arch 30 feet from the center (which is a point on the ellipse) is 11 feet, so we have $x = 30$ and $y = 11$. We can use these values to solve for $b$, the semi-minor axis.Plugging $x = 30$ and $a = 46$ into the ellipse equation, we get:$\frac{30^2}{46^2} + \frac{11^2}{b^2} = 1$Solving for $b^2$, we find the value of $b$, which represents half the total height of the arch at its center. However, since we are given that the height at 30 feet from the center is 11 feet, we are actually looking for the maximum height of the arch, which corresponds to $b$.After solving the equation, we find that the height of the arch at its center is approximately 14.51 feet, making option A the correct answer.

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