Chapter 9.4, Problem 27WE

To determine

To find: the height of the arch above the water

Answer to Problem 27WE

The height of the arch above the water is 3 m.

Explanation of Solution

Given:

The span of the bridge is 12m

Height is 4m

Calculation:

Let us consider a semi-ellipse with the major axis horizontal, focal radii $2a$ , and height b .

The span of the bridge is the length of the major axis, $2a$ .

  $\begin{array}{l}2a=12\\ a=\frac{12}{2}\\ =6\end{array}$

The height of the ellipse, b , is the height of the arch above the water at the center, which is 4 m.

Since the major axis is horizontal, the foci are on the x -axis.

The ellipse having center $\left(0,0\right)$ , foci $\left(-c,0\right)$ and $\left(c,0\right)$ , and sum of focal radii $2a$ has the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $b^2=a^2-c^2$ .

Substitute 6 for a , and 4 for b in the equation to get the equation of the ellipse.

  $\begin{array}{l}\frac{x^2}{6^2}+\frac{y^2}{4^2}=1\\ \frac{x^2}{36}+\frac{y^2}{16}=1\end{array}$

Now, consider a point on the major axis, 2m from one of the ends of the ellipse.

The distance of this point from the center of the ellipse is $\frac{12}{2}-2=4m$ .

Find the height of the arch from the point $\left(4,0\right)$ .

For this, consider a point vertically above $\left(4,0\right)$ on the arch.

Let h be the height of the arch above this point.

The coordinates of the point on the arch is $\left(4,h\right)$ .

Substitute 4 for x , and h for y in the equation of the ellipse and simplify.

  $\begin{array}{l}{h}^2=16\left(1-\frac{4}{9}\right)\\ =16-\frac{64}{9}\\ =\frac{144-64}{9}\\ =\frac{80}{9}\end{array}$

Take the square root on both the sides.

  $\begin{array}{l}\sqrt{h^2}=\sqrt{\frac{80}{9}}\\ h=\frac{4\sqrt{5}}{3}\approx 3\text{ m}\end{array}$

Conclusion:

Therefore, the height of the arch above the water is 3 m.