SOLUTION: A truck has to pass under an overhead parabolic arch bridge which has a span of 20 meters and is 16 meters high. If the tank is 14 meters wide, is placed in the truck with its sid

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A truck has to pass under an overhead parabolic arch bridge which has a span of 20 meters and is 16 meters high. If the tank is 14 meters wide, is placed in the truck with its sid      Log On


   

Question 1204361: A truck has to pass under an overhead parabolic arch bridge which has
a span of 20 meters and is 16 meters high. If the tank is 14 meters wide, is placed in the truck with its sides vertical, and the top of the tank is 7.5 meters above the street level, what is the smallest clearance from the top of the tank so that the truck can pass under the bridge?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let the origin be the point on the roadway directly below the top of the arch. Then the equation of the parabola is

y=ax%5E2%2B16

where a is a negative constant to be determined.

With the arch having a span of 20 meters, the coordinates of the two bases of the arch are (-10,0) and (10,0). Use one of them to determine the constant a.

0=a%2810%5E2%29%2B16
100a=-16
a=-0.16

The equation of the parabola is

y=-0.16x%5E2%2B16

The smallest clearance between the tank on the truck and the arch will be if the truck is centered under the arch. Since the tank is 14m wide, that clearance will be 7m from the center of the arch. Use the equation of the parabola to find the height of the arch 7m from the center.

y=-0.16%287%5E2%29%2B16=8.16

Then, since the top of the tank is 7.5m above the roadway, the clearance is 8.16m - 7.5m = 0.66m.

ANSWER: 0.66 meters