## Related questions with answers

Elliptical concrete pipes have a greater capacity for shallow flow than circular pipes. For this reason, elliptical pipes are often used in storm drains, culverts, and sewers. An engineer wants to design an elliptical concrete pipe with a maximum horizontal opening of 4 ft and a maximum vertical opening of 3 ft. a. Write an equation for an elliptical cross section of the pipe. For convenience, place the coordinate system with (0,0) at the center of the pipe. b. To construct the pipe, the engineer needs to know the location of the foci. How far from the center are the foci?

Solution

Verified$\textbf{a.}$

The ellipse is horizontal so we use the standard form:

$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$

The center is at the origin so $(h,k)=(0,0)$.

The length of the major axis is $2a$ and is the maximum horizontal opening of 4 ft:

$2a=4$

$a=2$

The length of the minor axis is $2b$ and is the maximum vertical opening of 3 ft:

$2b=3$

$b=1.5$

So, the equation is:

$\dfrac{(x-0)^2}{2^2}+\dfrac{(y-0)^2}{1.5^2}=1$

$\color{#c34632}\dfrac{x^2}{4}+\dfrac{y^2}{2.25}=1$

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