SOLUTION: A rectangular object 25 m wide is to pass under a parabolic arch that has a width of 32 m at the base and a height of 24 m at the center. If the vertex of the parabola is at the to

Question 1196473: A rectangular object 25 m wide is to pass under a parabolic arch that has a width of 32 m at the base and a height of 24 m at the center. If the vertex of the parabola is at the top of the arch, what maximum height should the rectangular object have?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The vertex is to be at the top of the arch; let the origin of a coordinate system be at the center of the base of the arch.

With a height of 24 at the center of the arch, the coordinates of the vertex are (0,24).

And with a width of 32 at the base, the coordinates of the endpoints of the arch are (-16,0) and (16,0).

With those conditions, the equation of the parabolic arch is , where a is a (negative) constant to be determined.

That constant can be determined using either endpoint of the arch.

The equation of the arch is

The object that is to pass through the arch has a width of 25, so it will extend 12.5 units each side of the center of the arch. To find the maximum height of the object, evaluate the equation for x=12.5. The answer is an ugly number, so I'll let you do that part. You might want to use a graphing calculator instead of working with the ugly arithmetic....

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