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 34 feet and the height of the arch over the center of the roadway is 12 feet. Two trucks plan to use
This road. They are both 8 feet wide. Truck 1 has an overall height of 12 feet and Truck 2 has an overall height of
11 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the bridge.
A)Neither Truck 1 nor Truck 2 can pass under the bridge.
B)Truck 1 can pass under the bridge, but Truck 2 cannot.
C)Truck 2 can pass under the bridge, but Truck 1 cannot.
D)Both Truck 1 and Truck 2 can pass under the bridge.

Question

Quizplus · Accepted Answer

First, we need to find the equation of the semi-elliptical arch. We know that the width of the roadway is 34 feet and the height of the arch over the center of the roadway is 12 feet. We can use this information to find the semi-minor axis of the ellipse (the distance from the center of the ellipse to the edge of the roadway):
b = 1/2(34/2) = 8.5 feet
We also know that the height of the arch over the center of the roadway is 12 feet, which gives us the semi-major axis of the ellipse:
a = b + 12 = 20.5 feet
So, the equation of the ellipse is:
(x^2/20.5^2) + (y^2/8.5^2) = 1
where x is the horizontal distance from the center of the ellipse and y is the vertical distance from the center of the ellipse.
Now, we can determine whether Truck 1 and Truck 2 can pass under the bridge. 
Truck 1 has an overall height of 12 feet, which means that it will hit the arch at its highest point. We can plug in x=0 into the equation of the ellipse to find the maximum height of the arch:
(0^2/20.5^2) + (y^2/8.5^2) = 1
y^2/8.5^2 = 1
y = +/- 8.5
So, the maximum height of the arch is 8.5 feet, which is less than the overall height of Truck 1. Therefore, Truck 1 cannot pass under the bridge. 
Truck 2 has an overall height of 11 feet. We can again plug in x=0 into the equation of the ellipse to find the maximum height of the arch:
(0^2/20.5^2) + (y^2/8.5^2) = 1
y^2/8.5^2 = 1
y = +/- 8.5
So, the maximum height of the arch is still 8.5 feet. However, since Truck 2 is only 11 feet tall, it can pass under the bridge. Therefore, the answer is C: Truck 2 can pass under the bridge, but Truck 1 cannot. To determine which truck can pass under the bridge, we need to compare the height of each truck to the height of the arch at the point where the truck will pass through.

Let's assume that the center of the arch coincides with the center of the roadway. Then, the distance from the center of the roadway to the closest point on the arch (the minor axis) is 17 feet, which is half of the width of the roadway.

For Truck 1, which is 12 feet tall, the distance from the center of the roadway to the top of the truck is 12/2 + 8 = 14 feet. This is greater than the height of the arch at the minor axis, which is 12 feet. Therefore, Truck 1 cannot pass under the bridge.

For Truck 2, which is 11 feet tall, the distance from the center of the roadway to the top of the truck is 11/2 + 8 = 12.5 feet. This is less than the height of the arch at the minor axis, which is 12 feet. Therefore, Truck 2 can pass under the bridge.

Hence, the answer is C, Truck 2 can pass under the bridge, but Truck 1 cannot.

Solve the Problem. -The Arch Beneath a Bridge Is Semi-Elliptical, a One-Way Roadway