Q. 27 - Precalculus Enhanced with Graphing Utilities

Question

Parabolic Arch Bridge A horizontal bridge is in the shape of a parabolic arch. Given the information shown in the figure, what is the height h of the arch 2 feet from shore? 

Step-by-Step Solution

Answer

The height is $3.6 \mathrm{unit}$.

Step 1. Given information.

Parabolic Arch Bridge A horizontal bridge is in the shape of a parabolic arch. Given the information shown in the figure, what is the height h of the arch 2 feet from shore? 

Step 2. Equation of parabola.

The goal of this task is to find the coordinates of the points on the parabola.

The equation of parabola is given by:

$y=a{(x-h)}^2+k$

where $\left(h,k\right)$ are the coordinates of the vertex.

Step 3. Find the height.

It is clear from the figure that the vertex has coordinates $\left(0,10\right)$, the point $\left(10,0\right)$ is on the parabola, too. Find $a$.

Here

$$\begin{gathered} \left(h,k\right)=\left(0,10\right) \\ \left(x,y\right)=\left(10,0\right) \end{gathered}$$

So,

 $$\begin{gathered} y=a{\left(x-h\right)}^2+k \\ 0=a{\left(10-0\right)}^2+10 \\ 0=100a+10 \\ a=-\frac{10}{100}=-0.1 \end{gathered}$$

Equation of parabola is:

$y=-0.1x^2+10$

To find y-coordinate of the point, put $x=8$ in the equation:

$$\begin{gathered} y=-0.1x^2+10 \\ y=-0.1{\left(8\right)}^2+10 \\ y=-0.1\left(64\right)+10 \\ y=-6.4+10 \\ y=3.6 \end{gathered}$$

The height is $3.6 \mathrm{unit}$.

Step 4. Conclusion.

The height is 3.6 unit.