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 30 feet and the height of the arch over the center of the roadway is 13 feet. Two trucks plan to
Use this road. They are both 10 feet wide. Truck 1 has an overall height of 12 feet and Truck 2 has an
Overall height of 13 feet. Draw a rough sketch of the situation and determine which of the trucks can pass
Under the bridge.
A)Truck 1 can pass under the bridge, but Truck 2 cannot.
B)Both Truck 1 and Truck 2 can pass under the bridge.
C)Neither Truck 1 nor Truck 2 can pass under the bridge.
D)Truck 2 can pass under the bridge, but Truck 1 cannot.

Question

Quizplus · Accepted Answer

The semi-elliptical arch has a major axis (width) of 30 feet and a minor axis (height) of 13 feet. The equation of an ellipse centered at the origin is $x^2/a^2 + y^2/b^2 = 1$, where 2a is the length of the major axis and 2b is the length of the minor axis. Here, $a = 15$ and $b = 13$. To determine if the trucks can pass, we need to check if their heights at their respective positions (5 feet from the center for each side of the truck, given they are 10 feet wide) are less than the height of the arch at those points. Truck 1, being shorter, can pass under the arch, but Truck 2, matching the maximum height of the arch, might not safely pass due to potential clearance issues.