If a trailer is 9m wide and stands 3.2m tall, measured from the ground to the top of the trailer, will it fit under the bridge (more info below)? A hauling company needs to determine whether a large...
If a trailer is 9m wide and stands 3.2m tall, measured from the ground to the top of the trailer, will it fit under the bridge (more info below)?
A hauling company needs to determine whether a large house trailer can be moved along a highway that passes under a bridge with an opening in the shape of a parabolic arc, 12m wide at the base and 6m high in the center. This is a quadratic application. Please include all steps, I don't understand how to do this problem.
We will assume that the arc is a parabola facing down.
Then the equation of the parabola would be:
f(x) = -ax^2 + bc + c
Now we will assume that the highest point is on the y-axis. Then the point ( 0,6) is the maximum point.
Now we will assume that the endpoints of the parabola is on the x-axis where the distance is 12 m.
Then the endpoints are ( 6, 0) and (-6, 0)
We will substitute and determine a, b , and c.
First we will substitute ( 0, 6)
==> f(0) = c = 6
==> c = 6
Now we will substitute 6 and -6/
==> f(6) = -36a + 6b + 6 = 0
==> -6a + b + 1 = 0 .............(1)
f(-6) = -36a - 6b + 6 = 0
==> -6a -b + 1 = 0............(2)
Now we will add (1) and (2).
==> -12a + 2 = 0
==> a = 1/6
==> b= -6a+ 1 = 0
==> b= 0
==> f(x) = -(1/6)x + 6
Now we need to determine if the trailer of 9 ft wide can fit under the bridge.
If the trailer drive exactly in the middle of the bridge then it will be between the points ( 4.5, 0) and ( -4.5, 0).
Now we need to find out the height of the bridge at these points.
We will substitute with f(4.5).
==>f(4.5) = -(1/6) (4.5)^2 + 6 = 2.635.
Then, the height of the bridge at the points (4.5, 0) is 2.635 m.
But the trailer height is 3.2
Then, the trailer can not fit under the bridge.