Solve Applied Problems Involving Ellipses
Solve the problem.
-The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch. The width of the roadway is 38 feet and the height of the arch over the center of the roadway is 11 feet. Two trucks plan to
Use this road. They are both 8 feet wide. Truck 1 has an overall height of 10 feet and Truck 2 has an overall
Height of 9 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the
Bridge.
A)Both Truck 1 and Truck 2 can pass under the bridge.
B)Neither Truck 1 nor Truck 2 can pass under the bridge.
C)Truck 1 can pass under the bridge, but Truck 2 cannot.
D)Truck 2 can pass under the bridge, but Truck 1 cannot.

Question

Quizplus · Accepted Answer

The semi-elliptical arch can be described by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $2a$ is the width of the roadway (38 feet) and $2b$ is the height of the arch (22 feet, but since it's semi-elliptical, we use 11 feet). Solving for $a$ and $b$, we find $a = 19$ and $b = 11$. Both trucks can pass under the bridge because their heights (10 feet and 9 feet) are less than the maximum height of the arch (11 feet), and their combined width (16 feet) is less than the width of the roadway (38 feet).