Question

Semi elliptical Arch Bridge An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river $20$meters wide. The center of the arch is $6$meters above the center of the river. See the figure. Write an equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch.

Short answer

The equation for the ellipse is$\frac{x^2}{100}+\frac{y^2}{36}=1$.

Step-by-step solution

Step 1. Given information.

From figure we have major axis equal to $20m$.

The major axis of an ellipse is $2a$.

So,

$$\begin{gathered} 2a=20 \\ a=10 \end{gathered}$$

From figure the semi minor axis is $b=6$

The x-xis coincide with the water level and y-axis passes through the center of the arch. The center of the ellipse is $(0,0)$

Step 2. Equation of the ellipse.

Substitute $a=10$and $b=6$in the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$we get.

$$\begin{gathered} \frac{x^2}{{10}^2}+\frac{y^2}{6^2}=1 \\ \frac{x^2}{100}+\frac{y^2}{36}=1 \end{gathered}$$