Question

. A bridge over a river is built in the shape of a parabolic arch. The bridge has a span of 100 feet. The height of the arch is 10 feet at a point 40 feet from the center. Find the height of the arch at its center.

Answer 1

The height of the arch at its center is 250/9 or about 27.78 feet.

Explanation:

We can write an equation to model the parabolic arch.

Let the left-most point of the arch be the origin (0, 0).

Since the bridge has a span of 100 feet, the right-most point must be (0, 100).

We can use the factored form of a quadratic:

`y=a(x-p)(x-q)`

Where p and q are the x-intercepts.

Our x-intercepts are x = 0 and x = 100. Hence:

`y=ax(x-100)`

At a point 40 feet from the center, the height of the arch is 10 feet.

The center is x = 50. So, a point 40 feet from the center can be either x = 10 or x = 90.

So, for instance, when x = 10, y = 10. Substitute and solve for a:

`10=10a(10-100)`

So:

`\displaystyle a=-(1)/(90)`

The same value will result if we let x = 90 and y = 10.

Hence, our equation is:

`\displaystyle y=-(1)/(90)x(x-100)`

The height of the arch at its center will be when x = 50. Hence:

`y(50)=\displaystyle -(1)/(90)(50)((50)-100)=(250)/(9)\approx 27.78\text{ feet}`