Question

A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 166 feet and a maximum height of 40 feet. Find the height of the arch at 10 feet from its center.

Answer 1

38.27775 feet

Explanation:

The bridge has been shown in the figure.

Let the highest point of the parabolic bridge (i.e. vertex of the parabola) be at the origin,
`O(0,0)` in the cartesian coordinate system.

As the bridge have the shape of an inverted parabola, so the standard equation, which describes the shape of the bridge is

`x^2=4ay\;\cdots(i)`

where
`a` is an arbitrary constant (distance between focus and vertex of the parabola).

The span of the bridge = 166 feet and

Maximum height of the bridge= 40 feet.

The coordinate where the bridge meets the base is
`A(83, -40)` and
`B(-83, -40).`

There is only one constant in the equation of the parabola, so, use either of one point to find the value of
`a`.

Putting
`A(83,-40)` in the equation (i) we have

`83^2=4a(-40)`

`\Rightarrow a=-43.05625`

So, on putting the value of
`a` in the equation (i), the equation of bridge is

`x^2=-172.225y`

From the figure, the distance from the center is measured along the x-axis, x coordinate at the distance of 10 feet is,
`x=\pm 10` feet, put this value in equation (i) to get the value of y.

`(\pm10)^2=-172.225y`

`\Rightarrow y=-1.72225` feet.

The point
`P_1(10,-1.72225)` and
`P_2(-10,-1.72225)` represent the point on the bridge at a distance of 10 feet from its center.

The distance of these points from the x-axis is
`d=1.72225` feet and the distance of the base of the bridge from the x-axis is
`h=40` feet.

Hence, height from the base of the bridge at 10 feet from its center

`= h-d`

`=40-1.72225=38.27775` feet.